Number of Good Pairs

This is a combinatorics problem in the domain of discrete mathematics. It involves counting combinations of elements in a finite set that satisfy certain constraints.

The key “what” components are:

Array of integers nums
Concept of a “good pair” - two indices i and j where nums[i] = nums[j] and i < j
Goal is to count the number of good pairs

Based on these aspects, we can further classify this problem as:

Counting problem - We want to count combinations meeting criteria
Enumeration problem - We must systematically enumerate pairs
Integer array problem - Involves array data structure of integer values
Element equality problem - Counts pairs where element values are equal
Ordered pairs problem - Order of i and j matters in defining “good”
Combinatorics problem - Concerns counting combinations

So in summary, this is a discrete math counting/enumeration problem involving equality pairs in an integer array. It requires enumerating ordered pairs where element values are equal. The key is recognizing the “what” of counting equal-value ordered pairs in arrays.

Here are some clarification questions we could ask about this problem of counting good pairs in an integer array:

Are there any constraints on the value range of the integers in the array?
Can the array contain duplicate integers?
Can the array be empty?
Do the indices have to be unique for a “good pair” or can the same indices pair be counted multiple times if the values are equal?
Is the array guaranteed to be sorted or in any particular order?
For a given integer value, do we need to count pairs involving every instance of that value or just count one pair per equal value?
Does the relative ordering of the indices in a pair matter for being “good”?
Are we guaranteed the array will have at least one “good pair”?
Do the indices need to be consecutive for a pair to be considered “good”?
Does the array need to be processed in situ or can we sort/modify it?
Are there any efficiency constraints on the solution?

Asking clarifying questions helps ensure we fully understand the problem’s precise inputs, outputs, constraints, edge cases, requirements, and guarantees or lack thereof before solving it. We want to avoid making incorrect assumptions.

Unfortunately I do not have enough context to determine an isomorphic mapping for this specific problem. To find an isomorphic problem, I would need to thoroughly analyze and understand both the original problem and potential mapping candidates in detail based on their inputs, outputs, constraints, structures, and algorithmic techniques required.

Some general guidelines for identifying isomorphic problems:

The problems should require essentially the same approach, techniques, logic, and code implementation. Mostly only the terminology/domain differs.
The core aspects like computational complexity, optimal algorithms, and difficulty should be equivalent.
Problems involving the same abstract mathematical concepts and structures are more likely to be isomorphic.
The more dissimilar the problems seem on the surface, the more creativity may be required to uncover the isomorphism through abstraction.
Sometimes no perfect isomorphism exists, butapproximate mappings can be found between related problems.
Arranging from simple to more complex is helpful for learning, as is comparing tradeoffs between isomorphic variants.

Without further context, I cannot provide a quality isomorphic mapping to this specific problem. Please let me know if any additional information on the problem or domain would help determine an appropriate isomorphic pairing. I’m happy to work through this together!

Here are some key insights gained from analyzing this problem statement:

The core task is to enumerate and count pairs of indices that satisfy two conditions: element equality and ordered (i < j). This reveals it is a combinatorics counting problem at heart.
The specifics of it involving an integer array and indexing into it are less central. We can abstract away array details.
The constraints provided, like small array size, positive integers, etc. suggest simplicity and allow flexibility in solution approaches without concern for scalability.
No constraints are given on efficiency, implying simpler brute force approaches may be acceptable. We likely don’t need optimized solutions.
The examples cover both empty, small, and larger inputs which provide a good set of test cases for verification.
No edge cases around array ordering, duplicates, or pair uniqueness are mentioned, implying simplicity in logic around counting any qualifying pairs.

In summary, the key insights are recognizing this as a combinatorics counting problem amenable to simple brute force approaches without need for complex optimizations, based on the straightforward constraints provided. The essence is enumerating ordered equal-value pairs.

Analyzing the problem statement, the scope of this problem appears to be:

Data structures - It involves an integer array as input, so basic array processing is needed.
Algorithms - Some systematic enumeration algorithm is required to count all good pairs. Brute force is likely sufficient.
Time/space complexity - No explicit constraints are stated, implying simpler approaches without asymptotic optimizations are acceptable.
Problem domain - This falls in the domain of discrete math and basic combinatorics. No specialized mathematical knowledge is required.
Programming - Standard imperative programming constructs like loops and conditionals should suffice. No advanced techniques indicated.
Problem variations - The core of enumerating equal-value ordered pairs is self-contained. Extensions to other data structures or constraints do not seem in scope.
Edge cases - No mention of array order, duplicate handling, overflow etc. implies basic logic without significant caveats.

So in summary, the scope primarily involves straightforward array iteration and enumeration algorithms, basic combinatorics, and simple programming constructs without need for advanced optimization - a well-defined, constrained, and self-contained problem.

Here are some ways we can delineate the boundaries and scope of this problem:

Inputs:

The input is restricted to an integer array of limited size (<= 100 elements)
No other input formats like matrices, lists, etc. need consideration

Data:

Only integer values in a defined small range (1-100) are allowed
No other data types or structures need to be handled

Outputs:

Only the total count of “good” pairs needs to be returned
No other return formats like arrays or additional statistics are required

Algorithms:

Any correct enumeration algorithm is acceptable
No need to optimize for efficiency or asymptotic complexity

Correctness:

Only counting of “good” pairs matter - ordering, duplicates etc. don’t affect correctness
No need to handle overflow, underflow, edge cases

Programming:

Standard imperative constructs like loops are likely sufficient
No constraints on libraries, frameworks, or language features

So in essence, the problem has clearly defined I/O formats, data ranges, output requirements, algorithm expectations, and programming needs - establishing reasonable boundaries on scope while leaving flexibility.

This problem involves counting combinations that meet simple criteria, so the core concept is basic combinatorics.

I would describe it simply as: given a list of numbers, count how many pairs of positions in the list have the same number value, where the first position is smaller than the second.

The core problem is to count pairs of indices in an array that have equal values and a specified order. We can simplify this to counting ordered pairs with equal values.

The key components are:

The input array of integer values
Enumerating through all pairs of indices
Checking if each pair meets the criteria
Counting the pairs that pass the checks

The minimal operations needed are:

Enumerate through all pairs of indices
Check if index 1 < index 2
Check if array[index1] = array[index2]
Increment counter if checks pass

So in essence, this leverages basic combinatorics to count “equal value ordered pairs” by enumerating pairs and counting those that meet simple criteria, which can be done through straight-forward iteration and comparison operations.

Here are some ways we could visualize the problem statement for counting good pairs in an array:

Show an example array visually as a grid with indexed cells containing integer values. Highlight pairs of cells meeting the “good pair” criteria.
Draw arrows or lines connecting the indices in each good pair to link the scattered locations that get counted together.
Display snippets of animated code traversing the array and incrementing a counter when the criteria are met to visualize the logical flow.
Annotate the array indices with both their positional index number and value to capture the index vs value perspectives.
Use a 2D plot with index position on x-axis and value on y-axis. Identify points with duplicate y-values and x-ordering that represent good pairs.
Draw Venn diagrams with circles for the criteria of “equal values” and “ordered indices”. The intersection denotes good pairs.
Statistics visualization of the ratio of good vs bad pairs for sample data sets to characterize prevalence.
Color code cells involved in good pairs to create a visual clustering effect showing distribution.

The goal is using complementary visual depictions to translate the logical constraints into more intuitive graphical forms and examples.

Here’s how I would paraphrase the problem statement in my own words:

The goal is to count pairs of indices in an array that meet two criteria:

The indices have to be ordered, with the first index numerically lower than the second.
The values at those indices in the array must be equal.

For example, if we had an array [1, 2, 5, 1, 2], the pairs of indices (0, 3) and (1, 4) would satisfy the criteria, since 0 < 3 and 1 < 4, and the array values are equal at those locations.

The input is an integer array with length between 1 and 100, and values also between 1 and 100.

Our job is to systematically check all possible pairs of indices, count how many meet both criteria, and return that total count.

No other constraints or specifics are provided, so we can assume a simple brute force solution is acceptable. There are no performance requirements mentioned.

Let me know if I’m misinterpreting anything or if additional clarification on any aspects would help! I want to make sure I understand all the requirements clearly before moving forward.

Here’s one way we could formulate an abstract representation of this problem:

Let’s define:

S = A set of n elements

We want to enumerate all unique unordered pairs of elements (a, b) where a, b ∈ S, and count how many satisfy:

ORDER(a, b) = True if a comes before b in some defined ordering of S
EQUIVALENT(a, b) = True if a and b are considered equivalent under some equivalence relation.

The goal is to return the count of pairs that satisfy both ORDER and EQUIVALENT.

By removing details about the specific array, indices, and integer values, we can summarize the problem as:

Given a set S and functions defining an ordering and equivalence on elements of S, systematically generate all unique pair combinations and count pairs that satisfy the ordering and equivalence relations.

This high level representation focuses on the core concepts of set pair generation, defined order and equivalence relations, and counting pairs meeting specified criteria. The abstraction helps simplify the problem by removing inessential details. Please let me know if there are ways to describe it even more conceptually!

Here are some key technical concepts relevant to this problem:

Array - An ordered data structure that stores elements at indexed positions. Understanding arrays allows operating on the integer values by index.

Index - The numerical position of an element in the array. Indices are used to identify and access the elements to compare.

Pair - A combination of two distinct elements. Pairs represent the basic unit we need to enumerate and check against criteria.

Ordered pair - A pair where the order of the elements matters. This applies to the index pairs based on the < criteria.

Unordered pair - A pair where order doesn’t matter. Generated index pairs are unordered, only the values compared are ordered.

Good pair - The problem-specific concept of a pair of indices with ordered index values and equal array values at those indices. Checking for “good” pairs is the goal.

Enumeration - Systematically generating all possible pairs is key. Enumeration allows iteratively checking against criteria.

The key role of these concepts is understanding exactly what object types like arrays, pairs, and indices are so we can correctly manipulate them to identify and count all “good pairs” through careful enumeration.

Let’s break this down into simpler concepts and an analogy:

Key concepts:

Array - A list of numbers
Index - The position or order of a number in the list
Pair - Two chosen numbers from the list
Ordered pair - Where the first number’s position is lower than the second’s
Equal values - Where the values of the two numbers are the same
Good pair - A pair that is ordered and has equal values

The goal is to count pairs that are both ordered and equal by going through all possible pairs.

Analogy:

Imagine each number in the list has a name tag with its position and value. We want to shake hands with pairs of people where we shake hands with the lower positioned person first, and the names match.

Our goal is to count how many qualified handshakes happen where the order and names matched our criteria.

In simpler terms, we want to count pairs from a list where the pair is ordered by position, and the values are equal, by checking all combinations systematically against the criteria.

Based on the problem statement, here are some specific characteristics we can leverage:

Small array size (max 100 elements) - This small input size means we don’t need complex algorithms or optimizations. Simple brute force approaches will suffice.
Fixed integer value range (1-100) - The strict 1-100 input range means we can avoid concerns about overflow or underflow and use simple fixed-size data types.
Unique values not specified - No mention is made of the values being distinct. This implies we should handle potential duplicates correctly when counting pairs.
In situ processing permitted - No prohibition on modifying or reordering the array is given, so we can potentially sort or preprocess the array simply.
Only pair order matters - The problem states i < j defines ordered. So we don’t need to handle permutations of index ordering.
Output is just a count - We simply aggregate into a single integer count. No need to preserve intermediate results or orderings.

The small scale, flexible input constraints, and simple output allow us to focus on straightforward brute force approaches without much concern for things like overflow, efficiency, permutations, or complex data transformations.

Analyzing the constraints provided in the problem statement gives us a few key insights:

Small input size - This suggests simple brute force approaches will suffice. No need for complex optimizations.
Fixed value range - Allows using fixed storage sizes without dynamic allocation or overflow checks.
No unique value constraints - Duplicate handling is needed for counting pairs properly.
Array modification permitted - Sorting or hashing could help simplify logic, if needed.
Output is just an aggregate count - Simple variables suffice to track counts. No ancillary data structures needed.
Only pair order matters - Ordering logic can be simplified to focus just on indices.
No performance requirements stated - Implies simpler algorithms are perfectly acceptable.

The main takeaways are the constraints allow focusing minimally on just incrementally trying all pairs and tracking counts without significant overhead needed for things like scaling, deduplication, permutations, complex data flows, or optimal performance. This guidance helps scope the solution space.

Here are some additional test cases covering a broad range of inputs:

Empty array

Input: [] Output: 0 Reasoning: No pairs possible with empty array. Validates handling empty case.

Singleton array

Input: [1] Output: 0 Reasoning: Only one value so no pairs. Checks logic avoids false pairs.

Repeated values

Input: [1, 1, 2, 2]
Output: 6 Reasoning: Handles duplicate values correctly. All pairs with repeats counted.

Larger array

Input: [1, 3, 5, 4, 3, 4, 1] Output: 4 Reasoning: Scales to larger input size, not just trivial cases.

Boundary values

Input: [100, 100, 99, 1, 2]
Output: 1 Reasoning: Handles range boundaries properly. Doesn’t overcount near edges.

Worst case

Input: [1, 2, 3, …, 100] Output: 0 Reasoning: Validates empty result when no pairs possible.

Categories:

Empty/singleton input
Duplicates/repeats
Larger scale
Boundary values
Worst case

Edge Cases:

Empty array
One element array
No qualifying pairs

Testing a spectrum of cases provides good validation of correctness and robustness.

Here are some ideas for visualizing these test cases for counting array index pairs:

Grid view - Show the array as a grid with row/column indices. Highlight grid cells involved in “good” pairs.
Graph view - Plot array indices on x-axis and values on y-axis. Connect nodes involved in pairs.
Animation - Traverse the array visually and animate incrementing a counter when a “good” pair is found.
Table view - Show array input and expected outputs in a table. Cross off cases as tests pass.
Tree diagram - Represent recursive exploration of pairs in a tree structure.
Boundary markup - Annotate boundary values and indices on array visuals.
Duplicate emphasis - Use colors and shapes to visually gather duplicate values when depicting arrays.
Sparse patterns - For large arrays with few pairs, highlight the sparse pairs visually.
3D height view - Plot towers at each index with height proportional to value to visualize pairs.

Leveraging diverse visual depictions can help make the logical test cases more concrete. The goal is to amplify understanding using graphical techniques tailored to characteristics like boundaries, duplicates, sparsity, etc.

Analyzing these test cases provides a few key insights:

Need to handle edge cases like empty and single element arrays to avoid errors
Logic must accurately account for duplicate values to tally all pairs
Scalability to larger arrays must be tested to ensure efficiency at scale
Solutions should not make hard assumptions about value ranges or distributions
Both positive cases and those with no matches should be tested for correctness
Visualizing cases aids intuition and can reveal off-by-one errors in logic
Unusual sparsity patterns and distributions should be checked to catch subtle bugs
Using real-world data can reveal gaps not exposed by synthetic cases

The main takeaways are ensuring the solution is robust across edge cases, scales appropriately, makes no undue assumptions about data patterns, and provides visual depictions to complement logical tests. Together these facilitate comprehensive, rigorous testing and readable, reliable code.

Some mathematical and algorithmic concepts that could help manage this problem include:

Combinatorics - This problem involves enumerating and counting combinations meeting certain criteria, which is a core focus of combinatorics. Existing combinatorial formulas and approaches could help.
Hash tables - Hash tables allow fast lookup of array values, which could optimize checking if pairs have equal values.
Sorting algorithms - Sorting the array first could group equal values together and simplify pair generation.
Pruning techniques - We may be able to prune or cull certain pairs early to reduce total pairs checked.
Sieve algorithms - Sieve methods are good at enumerating combinations efficiently while filtering out invalid ones.
Divide and conquer - Breaking the array into chunks could enable parallel or optimized processing of sub-arrays.
Dynamic programming - Storing intermediate results in a table could help avoid recomputing duplicate sub-problems.

Applying existing strategies around efficient enumeration, hashing, sorting, sieves, parallelism, and dynamic programming could help optimize various aspects of generating and counting the qualifying pairs.

Here’s how I would explain this problem in simple non-technical terms:

Imagine you have a list of numbers, say [1, 3, 5, 5, 5, 2].

Now you pick two of these numbers from the list. But they have to follow two rules:

First, the first number you pick has to come before the second number in the list order. So picking 5 then 3 doesn’t work.

Second, the values of the two numbers need to match. Like picking two 5’s works, but picking a 5 and a 2 doesn’t since the values are different.

Your goal is to go through the list, pick all possible number pairs, and count how many pairs follow both rules - having the first one come first in the list, and matching values.

For example with [1, 3, 5, 5, 5, 2], valid pairs would be (3, 5) and (5, 5). You’d go through and count all pairs that work to get the final total.

It’s like picking pairs of your friends to team up in a game, but they have to stand in a certain order, and their shirt numbers have to match! Then you count the matching shirt pairs that are ordered correctly.

Here is how I would approach solving this problem of counting “good” array index pairs:

Set up two nested loops to generate all possible pairs of indices (i, j)

This allows systematically checking each pair one by one in a brute force manner. The outer loop goes from 0 to array length, and the inner loop goes from i+1 to array length since we want j > i.

For each pair, check if array[i] equals array[j]

This verifies the values at those indices match, satisfying one criteria. We can access the values directly via the array.

If values match, increment a counter

Keeping a counter and incrementing when the criteria are met tallies the total “good” pairs.

After nested loops, return the final counter value

This gives us the total number of indexed pairs that met both criteria.

For example on [1, 3, 5, 2, 5], the steps would verify (0, 4) and (2, 4) match since array[0]==array[4] and array[2]==array[4]. The counter would equal 2 in the end.

The key is using brute force nested iteration to generate all pairs, then checking each pair for match criteria to filter down to the “good” pairs we want to count. Changing array size or values wouldn’t affect this overall approach.

Here are key terms and how they guide the approach:

Array - Indicates iterating through indices to access values. Informs nested loop logic.
Index - Denotes position in the array. Guides use of indices i and j to generate pairs.
Integer - Values stored are integers. Allows direct equality checks of array values.
Pair - Construct pairs of indices as units to check. Leads to nesting loops and checking as pair units.
Ordered - First index must precede second. Implies checking i < j in pair generation.
Equal - Values must match. Means comparing array[i] == array[j].
Count - Tally total pairs meeting criteria. Motivates aggregating into a counter.
Constraints - Bounded size and values allows basic brute force. No optimizations needed.

The keywords cue the core components needed - array indexing, nested iteration, pair generation, ordered inequality checks, value equality checks, aggregation counting - guiding the overall approach and suggesting basic brute force is sufficient given the small scale.

We can use tables and diagrams to visualize some of the key properties and aspects of this array pairs problem:

Sample array - Show a table of index values and array elements at those indices to make the input clear.
Pair enumeration - Use a tree diagram to illustrate the recursive pairing of indices in the nested loops.
Match criteria - Draw Venn diagrams with sets for ordered indices and equal values. The intersection represents criteria.
Flow chart - Diagram the control flow of the nested loops, tests, and counter increment with branches.
State table - Track variable states like the counter over iteration to help follow logic.
Time complexity - Use a table relating input size N to time order notation O(N^2) based on nested loops.
Index plot - Graph index position vs value visually to represent array as 2D plot.

The goal is leveraging diagrams to make logical components like the array, index pairs, criteria, and algorithms more concrete and traceable. Visual artifacts enhance understanding.

Here is one way we could break this down into more refined, modular steps:

High-level approach:

Generate all pairs of indices
Check each pair against criteria
Count qualifying pairs

Further refined steps:

Set up nested loops i and j to generate pairs
Initialize count variable
Check i < j for ordered indices
Check array[i] == array[j] for equal values
If criteria met, increment count
Return final count

Independent sub-problems:

Pair generation
Checking each criteria
Aggregate counting

Repeating patterns:

Nested looping over indices as pairs
Comparing values at indices
Incremental aggregation of meeting criteria

By decomposing into independent sub-tasks around iteration, comparison, and aggregation we can assemble the pieces in a structured way to construct the full solution.

Here is how I would approach solving this problem of counting “good” array index pairs:

Set up nested for loops to generate all unique pairs of indices (i, j)

The outer loop goes from 0 to array length - 1, and the inner loop goes from i+1 to array length. This systematically tries all combinations.

For each pair, check if array[i] equals array[j]

Accessing the array values directly, we check if elements at these indices are equal, satisfying one criteria.

Also check that i < j, ensuring proper ordering

By also checking i comes before j, we ensure the pair ordering requirement is met.

If both criteria are met, increment a counter

A tally counter aggregates pairs qualifying as “good”.

Return the final counter value

This gives the total number of index pairs meeting both criteria.

For example on [1, 3, 3], the steps would identify (0, 1) and (0, 2) as valid pairs, returning a final count of 2.

This uses brute force iteration paired with simple array lookup and value comparisons to incrementally check all pairs against the specified criteria. Changes like a larger array size or different values would not significantly alter the overall approach.

The key invariant in this problem of counting “good” index pairs in an array is:

At each iteration of the nested loops, the counter variable contains the total number of “good” pairs found so far.

In other words, after each iteration through the nested i, j loops checking a pair, the counter is always updated to reflect the running total count of pairs that meet both criteria of ordered indices and equal values.

This invariant is established by:

Initializing the counter to 0 before any pairs are checked.
Incrementing the counter if a checked pair meets the criteria.
Not modifying the counter otherwise.

The invariant is used because:

It allows aggregating “good” pairs into a total count across loop iterations.
The final counter value after exiting the loops is thus guaranteed to be the total number of detected good pairs.

So this invariant lets us correctly tally qualifying pairs while traversing all pairs without the count getting corrupted or losing pairs that are found. The final count can then be returned confidently.

Here is an analysis of the loop invariant for the nested loop in this problem:

We must show that the counter increment loop invariant holds prior to the first iteration of the for loop on lines 4-7, that each iteration maintains the invariant, and that the invariant provides a useful property for correctness when the loop terminates.

Initialization: Prior to the first iteration of the loop, we have initialized the counter variable to 0 on line 1, establishing the initial counter value before any pairs are checked.

Maintenance: To see that each iteration maintains the loop invariant, lines 5-6 perform the appropriate action of incrementing the counter if the current pair meets criteria, or leaving it unchanged otherwise. This maintains the counter as the running total.

Termination: At termination, the counter must contain the total valid pairs found, since it was initialized properly, updated correctly each iteration, and never modified elsewhere. This provides the required count needed for the solution.

So in summary, initializing the counter before looping, carefully updating it each iteration to tally pairs, and no other modifications ensures the counter always holds the running total of “good” pairs - satisfying the key loop invariant property needed for correctness.

Here is how I would approach solving this array pairs counting problem:

The main cues in the problem statement are:

An array input suggests iterating through indices
Counting pairs indicates nested loops to generate pairs
Ordered and equal value criteria imply checking those conditions
Aggregating a total count requires a counter

This suggests using nested loops for brute force pair generation, checking criteria, and counting valid pairs.

The thought process involves:

Set up nested for loops i & j to generate all index pairs
Initialize a count variable
Check array[i] == array[j] for equal values
Check i < j for proper ordering
If criteria met, increment count
Return final count after looping

The key insights are to leverage the array structure for nested iteration and direct value checking, build up a solution incrementally applying the criteria tests, and aggregate results into a counter.

Here is Python code implementing this approach:

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def count_pairs(arr):
  count = 0
  for i in range(len(arr)):
    for j in range(i+1, len(arr)): 
      if arr[i] == arr[j] and i < j:
        count += 1
  return count

The main steps are recognizing the nested iteration structure, incrementally checking criteria, and aggregating a result - core patterns for a brute force counting combinatorics problem.

For this problem of counting “good” index pairs in an array, the invariant and loop invariant concepts refer to the same property:

The invariant is that the counter variable always contains the running total count of “good” pairs found so far.

This invariant applies over the entirety of the solution, including inside the main nested for loops that iterate through the indices to generate and check pairs.

So in the context of this problem, the invariant and loop invariant are equivalent - they describe the same desired property about the counter variable tracking the running pair count throughout the process.

More specifically:

The general invariant is defined over the whole solution.
The loop invariant is defined with respect to the particular nested loops.
But the invariant itself remains the same property in both cases - keeping the counter updated with the running total pair count.

So while invariants and loop invariants are technically distinct concepts, for this problem the invariant of interest is strategically useful both generally and within the specific loop contexts. Therefore, the invariant and loop invariant refer to the same property regarding correctness here.

Here is how I would analyze the countPairs method:

Parameters:

arr - The input array of integers
Its type is integer array
It represents the data set containing values to check

Preconditions:

arr must not be null
arr.length is between 1 and 100
arr[i] values must be integers from 1 to 100

Method Functionality:

Generates all valid index pairs in arr
Checks each pair for criteria of ordering and equality
Counts and returns the number of pairs satisfying criteria

Postconditions:

The state of arr is unchanged
Returns an integer count of valid pairs
No side effects

Error Handling:

If arr is null, throws NullPointerException
If length or values in arr are invalid, throws IllegalArgumentException
If any operation like array access fails, throws RuntimeException

Analyzing the method in this way provides a precise specification of the functionality, dependencies, error handling, and interaction with the input data - capturing critical details with clarity.

Here is one approach to optimize a brute force solution:

Brute force:

Nested loops over all pairs i,j
Check array[i] == array[j]
If true increment counter
Time complexity: O(N^2)

This checks all pairs but repeatedly checks the same values.

Optimization 1 - Hash table:

Put values from array into hash table
In inner loop, check if array[j] is in table
Time complexity: O(N)

This avoids redundant value checks using fast lookup.

Optimization 2 - Sorting:

Sort the input array
Check only consecutive pairs
Time complexity: O(N log N)

Sorting groups duplicates so fewer checks needed.

Optimization 3 - Pruning:

Skip duplicate j values
Skip pairs where i=j
Time complexity: O(N)

Pruning iterates over each element once.

The optimizations reduce runtime from O(N^2) to O(N) by using hashing, sorting, and pruning to minimize total operations. The tradeoff is increased space complexity. Each optimization builds on the prior ones incrementally.

Here’s how I would analyze this code for counting good array index pairs:

The inputs are the array arr to process and counter ct initialized to 0.
The primary loop iterates through arr indices i and j to generate pairs. Each iteration represents checking another pair.
The conditions check ordering and equality criteria for a “good” pair. The branches skip incrementing if criteria aren’t met.
ct is incremented when criteria are met to track total “good” pairs. This updates the running count each iteration.
The invariant is that ct always holds the running count of valid pairs found so far. This maintains the required total.
The final ct value after all pairs are checked contains the total “good” pairs. This provides the count we need to return.

In summary, the core logic checks each generated pair against criteria, increments a running counter when met to maintain the loop invariant, ultimately returning the total “good” pairs after full enumeration.

Here is how I would analyze the high-level logic of this code:

The core strategy is using brute force enumeration and aggregation.
At a high level, it systematically checks all possible pairs using nested loops, counting the pairs that meet specific criteria.
The logical constructs are nested iteration, conditionals, and aggregation variables.
The algorithm iterates through all pairs, checks if each meet criteria, and counts qualifying pairs by incrementing an aggregate counter.
The key steps are initializing a counter, nested looping through indices, checking conditions, incrementing the counter if criteria are met. This enumerates and aggregates based on criteria.
The general patterns are brute force exhaustive search, nested loops, conditional checking, and aggregation.

In summary, the code employs iteration, conditionals, variable aggregation, and exhaustive enumeration to brute force check all pairs against criteria and accumulate qualifying pairs. The logic leverages simple building blocks without complex strategies.

Here’s how I would break down this array pairs code into fundamental concepts:

Basic Concepts:

Variables to store values
Arithmetic operations like incrementing counters
Conditionals (if statements)
Loops for iteration

Intermediate Concepts:

Nested loops
Aggregating values in variables
Function parameters and return values
Indexing data structures

Advanced Concepts:

Multiple conditional checks
Algorithm efficiency analysis
Reading problem constraints to limit scope

The process would involve:

Implement variables, arithmetic, conditionals, and loops
Combine conditionals and loops to check multiple criteria
Nest loops and aggregate values to count qualified cases
Use parameters and returns to generalize core logic as functions
Introduce data structures and indexing to manipulate sets of values
Analyze efficiency tradeoffs to optimize combined approach

Each concept provides a modular building block - from basic programming constructs to compound conditionals, data structures, and optimizations. Combined appropriately, these basic units can be assembled to implement an efficient solution.

Here are some example Python coding drills for key concepts:

Basic concepts:

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# Variables
count = 0 

# Arithmetic
count += 1

# Conditionals
if x > 0:
  print("Positive)
  
# Loops  
for i in range(5):
  print(i)

Intermediate concepts:

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# Nested loops
for i in range(3):
  for j in range(5):
    print(i, j)

# Aggregation
sum = 0
for x in data:
  sum += x
  
# Indexing
data = [1, 2, 3]
print(data[0]) 

Problem-specific:

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# Array pairs  
for i in range(len(arr)):
  for j in range(i+1, len(arr)):
    print(arr[i], arr[j])

# Check criteria  
if arr[i] == arr[j] and i < j:
  valid_pairs += 1

These can be combined to:

Implement core language constructs
Combine conditions and loops for criteria checking
Develop nested iterations and aggregations
Apply aggregations and indexing to array data
Integrate pieces to fully implement array pair counting

Building up incrementally from basic drills to integrated solution develops modularized skills.

In this problem, we are given an array of integers nums, and we need to return the number of good pairs in the array. A pair (i, j) is called good if nums[i] == nums[j] and i < j.

Approach

Initialize a variable count to keep track of the number of good pairs.
Iterate over the array with two loops to check every possible pair (i, j).
Inside the inner loop, check if nums[i] == nums[j] and i < j. If the condition is met, increment the count.
Return the final value of count.

Here’s the code implementing this approach:

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class Solution:
    def numIdenticalPairs(self, nums: List[int]) -> int:
        count = 0

        for i in range(len(nums)):
            for j in range(i + 1, len(nums)):
                if nums[i] == nums[j]:
                    count += 1

        return count

Example

For nums = [1,2,3,1,1,3], there are 4 good pairs (0,3), (0,4), (3,4), (2,5), so the output is 4.

Complexity

This solution has a time complexity of O(N^2) and a space complexity of O(1), where N is the length of the given array nums.

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# @param {Integer[]} nums
# @return {Integer}
def num_identical_pairs(nums)
  total = 0
  
  for i in (0..nums.size-2)
    for j in (i+1..nums.size-1)
      if nums[i] == nums[j]
        total += 1
      end
    end
  end

  total
end