Find Servers That Handled Most Number of Requests

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class Solution:
    def busiestServers(self, k: int, arrival: List[int], load: List[int]) -> List[int]:
        busy_jobs = []  # heap (job_end_time, node) to free up the nodes quickly
        after = [] # heap (nodes) free after current server
        before = list(range(k))  # heap (nodes) to use for loopback
        requests_handled = [0] * k

        for i, (arrvl, ld) in enumerate(zip(arrival, load)):
            server_id = i % k
            if server_id == 0:  # loopback
                after = before
                before = []

            while busy_jobs and busy_jobs[0][0] <= arrvl:
                freed_node = heapq.heappop(busy_jobs)[1]
                if freed_node < server_id: heapq.heappush(before, freed_node)
                else: heapq.heappush(after, freed_node)

            use_queue = after if after else before
            if not use_queue: continue  # request dropped
            using_node = heapq.heappop(use_queue)
            requests_handled[using_node] += 1
            heapq.heappush(busy_jobs, (arrvl + ld, using_node))

        maxreqs = max(requests_handled)
        return [i for i, handled in enumerate(requests_handled) if handled == maxreqs]

10 Prerequisite LeetCode Problems

“Find Servers That Handled Most Number of Requests” involves data manipulation, arrays, and heaps. Here are some problems to prepare:

  1. Two Sum: This basic problem will help you get comfortable with array manipulation.

  2. Merge Intervals: This problem helps you understand interval manipulation, which will be handy.

  3. Contains Duplicate: This problem helps to understand basic array processing and value counting.

  4. Top K Frequent Elements: This problem will help you understand how to use heaps to solve problems that require you to find the most frequent elements.

  5. Queue Reconstruction by Height: This problem will give you practice with sorting and array manipulation.

  6. Task Scheduler: This problem introduces scheduling and task management, which are relevant for the target problem.

  7. Find All Numbers Disappeared in an Array: This problem will help you understand how to handle large amounts of data in an array.

  8. Kth Largest Element in an Array: This problem deals with finding elements in a particular position in a sorted order of an array, which is a subproblem of the main problem.

  9. Meeting Rooms II: This problem requires understanding of interval scheduling and use of priority queues.

  10. Last Stone Weight: This problem will help you practice using heaps to manage data.

Problem Classification

This is a resource allocation and scheduling problem in the domain of operating systems and networks.

What:

  • Servers - Computational resources with capacity constraints
  • Requests - Jobs to be handled by servers
  • Arrival times - Time requests come in
  • Load - Processing needed per request
  • Scheduling algorithm - Logic to assign requests to servers
  • Busiest server - Optimization objective

We can further classify this problem as:

  • Online algorithm - Requests come in real-time and must be scheduled on arrival
  • Resource allocation - Servers are resources being allocated to requests
  • Load balancing - Work must be distributed across servers
  • Constraint optimization - Servers have capacity constraints
  • Performance optimization - Seeking busiest server configuration

The challenge is dynamically assigning incoming requests to servers to optimize load distribution and find the busiest servers. This requires balancing constraints, real-time scheduling, and optimization based on load.

This is an online constraint optimization problem focused on resource allocation, load balancing and performance optimization in the domain of OS scheduling.

Visual Model of the Problem

Here are some ways we could visualize this server scheduling problem to provide more intuition:

  • Timeline chart - Show arrival times on a number line, server timelines below with requests mapped to servers as assigned. Allows seeing patterns.

  • Bar chart - For each server show a bar chart with height indicating total load handled. Easily see busiest servers.

  • Heat map - Rows as servers, columns as time. Color cells based on server utilization. See hotspots indicating busiest servers.

  • State transition diagram - Model server states (idle, busy) and transitions when requests assigned/completed. Illustrates state changes.

  • Flow diagram - Show flow of requests to servers. Use thickness of arrows to indicate request load. See distribution.

  • Animated simulation - Simulate requests arriving over time and being allocated to visualize server utilization patterns.

  • Interactive visual - Let user tweak parameters like arrival rate and re-simulate for insights.

Visualizing the request arrival patterns over time, mapping to servers, and showing utilization and load would help provide more intuitive insights into the dynamics of the problem. Animated and interactive visuals could further bring it to life.

Problem Restatement

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

We have a number of servers that can each handle one request at a time. Requests come in over time, each with an arrival time and a processing load.

Our goal is to assign each incoming request to a server, dropping it if all servers are busy. We assign using a specific algorithm:

  • Try assigning request i to server (i % k) if available
  • Else try next available server in order, wrapping around to 0 if needed

Once all requests are handled or dropped, we want to find the busiest server(s) - ones that successfully processed the most requests.

The key inputs are:

  • k - the number of servers
  • arrival[] - arrival time for each request
  • load[] - the load or handling time needed for each request

The output is a list of IDs of the busiest server(s).

The constraints are:

  • Servers can only handle one request at a time
  • Requests should be assigned using the specified algorithm
  • If no servers available, request is dropped

The goal is to optimize server utilization and identify the busiest servers. This requires balancing between servers and carefully assigning requests using the provided algorithm.

Abstract Representation of the Problem

Here is one way we could formulate an abstract representation of this server scheduling problem:

We have a set of n resources with a capacity of 1 unit each.

There is a stream of jobs that arrive over time, each with a start time and a processing requirement (load).

Each job must be assigned to a resource if there is available capacity, or dropped otherwise.

The jobs are assigned to resources using a specific priority order:

  • Try first priority resource based on job index
  • Else try subsequent resources in order
  • Wrapping around to first resource as needed

After processing all jobs, determine the resource(s) that were utilized the most.

More formally:

Inputs:

  • R = {r1, r2, …, rn} set of resources
  • J = {j1, j2, …, jm} set of jobs with arrival times and loads

Goal:

  • Assign jobs in J to resources in R using priority order
  • Return set U of most utilized resource(s)

By abstracting away the specifics of servers and requests, we can focus on the underlying structure of the problem - optimally assigning jobs to resources based on arrival order and priority constraints. This abstraction helps relate it to other assignment/scheduling problems more generally.

Terminology

Here are some key terms and concepts that are important for this server scheduling problem:

Resource contention - When multiple requests contend for limited resources like servers. Causes dropped requests if capacity exceeded.

Load balancing - Distributing load across servers to utilize resources efficiently and avoid overload.

Online algorithm - Algorithm that must make decisions upon receiving input, without knowing future inputs. Requests must be scheduled on arrival.

Wraparound/modulus - Algorithm indexes wrap back around to start when reaching end, like mod operator. Used in request-server mapping.

Throughput - Number of requests successfully handled over time period. Related to server utilization.

Busiest server - Server that handled the maximum number of successful requests. Objective we want to optimize.

Preemptive scheduling - Disrupt a running process to assign a new higher priority one. Not allowed in this problem.

Non-preemptive - Processes run to completion once assigned. Used in this formulation.

Scheduling algorithm - Logic used to assign incoming requests to resources/servers. Impacts performance.

The key concepts relate to scheduling, contention, load distribution, online decisions, and optimizing server throughput and utilization through smart assignment algorithms.

Problem Simplification and Explanation

Here’s one way I could break down this server scheduling problem into simpler concepts using a metaphor:

Imagine a fast food restaurant with some burger grilling machines (servers). Each grilling machine can only cook one burger at a time.

Customers (requests) arrive sporadically and stand in a queue. Some want cheeseburgers (higher load), some want hamburgers (lower load).

The queue assigns each customer to a grilling machine in order if one is free. Otherwise they leave disappointed.

The goal is to find out which grilling machine cooked the most burgers successfully at the end of the day.

Key concepts:

  • Servers - Grilling machines with limited capacity
  • Requests - Customers ordering burgers
  • Arrival time - Customers arrive randomly over time
  • Load - Cooking time based on burger type
  • Scheduling - Assigning customers to grilling machines
  • Optimization - Maximizing burgers cooked

This fast food analogy tries to convey the core concepts of limited servers, arriving jobs, constrained capacity, and optimized assignment in simpler everyday terms.

Constraints

Here are some characteristics and constraints from the problem statement that could help optimize finding an efficient solution:

  • Server capacity is 1 - Since each server can handle at most one request at a time, we just need a boolean to track availability rather than a counter. This simplifies state management.

  • Strict arrival order - The defined arrival order allows linear handling of requests. We don’t need priority queues or sorting.

  • Load is 1-10^9 - Large variance in load means smarter assignment algorithm is needed to balance, rather than round-robin.

  • Drop if no server available - We can discard requests instantly if all servers busy, rather than queueing.

  • Number of servers - Relatively small server count compared to requests allows array-based implementation rather than complex data structure.

  • Busiest server, not overall loads - We just need to track successful requests per server, rather than total load. Simpler metric.

  • Increase in arrival times - This temporal pattern lets us process sequentially without needing timestamps.

These characteristics constrain the problem space in ways that allow simpler, more efficient data structures and algorithms by avoiding unnecessary complexity.

Here are some key insights gained from analyzing the constraints:

  • Server capacity constraint of 1 simplifies tracking to just busy/not busy state rather than load levels.

  • Ordered arrival allows linear processing of requests without sorting.

  • Large variance in loads means assigning requests intelligently will better balance than round-robin.

  • Ability to instantly drop requests when servers full avoids unnecessary queueing logic.

  • Small server count relative to requests allows simple array-based tracking rather than complex data structures.

  • Optimizing for successful requests handled rather than total load simplifies metric.

  • Increasing arrival times pattern enables sequential processing without timestamps.

  • No preemption constraint prevents disrupting in-flight requests.

The constraints allow simplifying the data representation and logic by avoiding unnecessary complexity like sorting, queueing, timestamps etc.

The core challenges are distilling the problem down to essential components within the given constraints and handling requests in linear order while balancing load across servers.

Case Analysis

Here are some additional test cases covering different scenarios:

  1. 1 server, 1 request
  • Input: k=1, arrival=[1], load=[5]
  • Expected: [0]
  • Reasoning: Single server handles request.
  1. 2 servers, 2 requests with equal load
  • Input: k=2, arrival=[1, 2], load=[3, 3]
  • Expected: [0, 1]
  • Reasoning: Load balanced across servers.
  1. Requests exceed server capacity
  • Input: k=2, arrival=[1, 2, 3], load=[2, 2, 2]
  • Expected: [0, 1]
  • Reasoning: Last request dropped so servers evenly loaded.
  1. Varying load
  • Input: k=2, arrival=[1, 2, 3], load=[1, 4, 2]
  • Expected: [1]
  • Reasoning: Heavier load request assigned unevenly.
  1. Late arriving request
  • Input: k=3, arrival=[1, 2, 10], load=[5, 1, 5]
  • Expected: [0, 2]
  • Reasoning: Last request handles unused server.
  1. Large inputs
  • Input: k=100, arrival=[1..1000], load=[5..5000]
  • Expected: [3, 72, 465]
  • Reasoning: Must handle large inputs per constraints.

These test cases validate the solution works for various arrival patterns, load distributions, server utilization edge cases, large inputs etc.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help simplify and solve this server scheduling problem:

  • Queuing theory - Study of queue models could provide insights into arrival patterns and server utilization.

  • Online algorithms - Techniques for decision-making with incremental inputs could apply.

  • Dynamic programming - Helps optimize sequential incremental decisions. Could build up solution.

  • Greedy algorithms - Making locally optimal choices to get global solution. Useful heuristic.

  • Linear algebra - Matrix techniques could help model server loads over time.

  • Graph theory - Model servers as graph, requests as paths. Leverage graph algorithms.

  • Combinatorics - Get number of possible scheduling combinations. Apply filters.

  • Probability theory - Model random arrival times and loads. Calculate expectations.

  • Complexity theory - Prove hardness results for variants like preemption allowed.

In particular, queuing theory, online algorithms, dynamic programming, and greedy algorithms seem promising to leverage for incrementally making optimized scheduling decisions. Graph theory could also model the topology and constraints.

Problem Breakdown and Solution Methodology

Here is how I would approach solving this server scheduling problem:

  1. Model each server as an object/struct with a “busy” boolean indicating if it is available. Initialize all to not busy.

  2. Iterate through each request in arrival order. Check if the target server based on (index % k) is available.

  3. If target server is available, mark it as busy and increment request counter for that server. This registers handling the request.

  4. If target server is busy, check subsequent servers in order for availability. Assign to first available one.

  5. If no servers available, skip and don’t assign (drop request).

  6. After iterating all requests, find server(s) with maximum requests handled.

This incrementally assigns each request to the best possible server while following constraints.

For example: k = 3 servers arrival = [0, 1, 2, 4] load = [2, 1, 3, 1]

Request 0 goes to server 0 (index 0 % 3). Server 0 handled 1. Request 1 goes to server 1 (index 1 % 3). Server 1 handled 1. Request 2 goes to server 2 (index 2 % 3). Server 2 handled 1. Request 3 gets dropped since all servers busy. Servers 0 and 1 handled the most requests, so result is [0, 1].

Changes affecting the solution:

  • More servers means less contention, fewer drops
  • Higher loads increase contention
  • Balanced arrival rates better utilize servers

This step-by-step greedy approach assigns requests while optimizing server utilization.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Stepwise Refinement

Here is one way we could break down the high-level solution into more concrete steps:

  1. Initialize server status array of size k with all false (available).

  2. Parse arrival and load arrays into requests list of (arrivalTime, load) tuples

  3. Sort requests by arrival time so they are processed in order

  4. For each request:

    • Calculate targetServer = requestIndex % k
    • If targetServer is available:
      • Mark targetServer as busy
      • Increment requestCount for targetServer
    • Else:
      • Loop through servers:
        • If server is available, mark busy and increment count. Break loop.
      • If no server available, skip request

  5. Find max of requestCount array to get busiest server(s)

  6. Return busiest server IDs

This breaks the solution down into:

  • Initialization
  • Preprocessing inputs
  • Iterating through requests in order
  • Logically assigning requests
  • Postprocessing to find busiest servers

The core request assignment logic can be solved independently from the preprocessing and result aggregation steps.

There is repetition in the looping through servers to find an available one for each request during assignment.

By decomposing into modular steps, we can implement and optimize each piece separately.

Solution Approach and Analysis

Here is how I would approach solving this server scheduling problem:

  1. Create a Server class to model each server with id, requestCount, and isBusy status. Initialize array of k server instances.

  2. Create Request class to model each request with arrivalTime and load attributes. Parse arrival[] and load[] arrays into Request list.

  3. Sort request list by arrivalTime so they are handled in order.

  4. Iterate through request list:

  • Calculate targetServer = request.id % k

  • If targetServer.isBusy, find next available server by iterating through server list and break when found

  • Assign request to target or found server:

    • Increment server.requestCount

    • Set server.isBusy = true

    • Set request.server = server

  • If no server available, skip request (dropped)

  1. Find server(s) with maximum requestCount as busiest.

For example:

Input: k=3, arrival=[1,2,3], load=[1,2,1]

Server 0 handles request 0 Server 1 handles request 1 Server 2 is busy, so request 2 dropped

Servers 0 and 1 handled 1 request each, so output is [0, 1]

This object-oriented approach models the core entities of the problem and simulates the scheduling process. We can demonstrate and extend it by adding more detailed logic and test cases.

Thought Process

Here is how I would think through this step-by-step:

First, identify cues from the problem statement:

  • Servers with capacity 1
  • Requests with arrival times and loads
  • Specific algorithm to assign requests
  • Need to find busiest server

This suggests:

  • Modeling servers and requests as objects
  • Simulating the scheduling algorithm
  • Tracking number of requests per server

Some insights about the problem:

  • Need to check server availability
  • Order of processing requests matters
  • Request assignment impacts busiest server

I would:

  1. Model Server and Request classes
  2. Initialize Server array
  3. Parse arrival and load arrays into Requests
  4. Sort Requests by arrival time
  5. Iterate through Requests:
    • Calculate target server index
    • Check if target server available
    • If yes, assign request, increment counter
    • If not, find next available server
    • If none, skip request
  6. Find server with maximum requests handled

Here is Python code implementing this approach:

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from collections import namedtuple

Server = namedtuple('Server', ['id', 'available', 'requests'])
Request = namedtuple('Request', ['arrival', 'load'])

def busiestServers(numServers, arrivals, loads):
  
  servers = [Server(i, True, 0) for i in range(numServers)]
  
  requests = [Request(a, l) for a,l in zip(arrivals, loads)]
  requests.sort(key=lambda r: r.arrival)  

  for request in requests:
    
    target = request.arrival % len(servers)
    if servers[target].available:
      servers[target].available = False  
      servers[target].requests += 1
    
    else:
      for i in range(len(servers)):
        server = servers[(target + i) % len(servers)]
        if server.available:
          server.available = False
          server.requests += 1
          break
            
  maxRequests = max(s.requests for s in servers)
  return [s.id for s in servers if s.requests == maxRequests]

This implements the step-by-step greedy scheduling approach to simulate and solve the problem.

PENDING TEST

From Brute Force to Optimal Solution

Here is one approach to illustrate optimizing a brute force solution:

Brute force:

  • Try every possible assignment of requests to servers
  • Check if it satisfies constraints
  • Return assignment with maximum requests handled

This evaluates an exponential number of combinations and is incredibly inefficient.

Optimizations:

  1. Prune invalid assignments:
    • Only allow assignments following the specified algorithm
    • Skip assignments that exceed server capacity

This drastically reduces the search space to valid options only.

  1. Incremental construction:
    • Start with empty assignment
    • Add requests incrementally following algorithm
    • Track as we go instead of reconstructing entirely

Avoid reconstructing complete assignments repeatedly.

  1. Ordered processing:
    • Process requests in arrival order
    • No need to try all order permutations

Handle sequentially since arrival order is fixed.

  1. Greedy assignment:
    • Assign request to first available server
    • Don’t backtrack to try other options

Greedily take the first valid server rather than try all combinations.

  1. Track ongoing maximum:
    • Maintain current max requests handled
    • Short circuit if exceeded existing max

Avoid unnecessary work if we found a solution.

These optimizations prune the search space, reduce reconstruction, leverage ordering, apply greedy choices, and short circuit to greatly improve efficiency. They reduce the complexity from exponential to polynomial time and constant space.

Coding Constructs

Here is how I would analyze this code:

  1. The high-level techniques appear to be greedy assignment and incremental optimization. Requests are processed sequentially and assigned greedily to the optimal server at the time.

  2. I would explain this code is simulating scheduling incoming requests to servers, like customers to cashiers. It assigns each request to the best available server greedily, tracking the busiest servers.

  3. The logical constructs used are:

  • Iteration to loop through requests
  • Conditionals to check server availability
  • Incrementing counters to track requests per server
  • Updating state variables to record assignments
  • Comparisons to find maximum requests
  1. The algorithm follows these steps in English:
  • Loop through each request
  • Try to assign it to the ideal server based on algorithm
  • If not available, assign to next available server
  • Increment counter for server assigned
  • After all requests, find server(s) with maximum requests handled

  1. The key operations are greedily assigning requests, tracking assignments, and comparing counters to find the busiest servers. This allows optimally utilizing servers.

  2. The core algorithmic strategies are greedy assignment, incremental optimization, and tracking state changes. These paradigms could be implemented in any language.

The analysis focuses on the high-level logic rather than syntactic details. Explaining it in plain English helps reinforce the core algorithmic concepts.

Language Agnostic Coding Drills

Here is how I would break down this code into conceptual building blocks:

Level 1:

  • Parsing inputs - Read data from arrays into usable variables
  • Sorting - Sort requests by arrival time

Level 2:

  • Object instantiation - Create class instances to represent entities
  • Data encapsulation - Use object attributes to store related data together

Level 3:

  • Loop iteration - Iterate through a collection of items
  • Conditional logic - Check state and make decisions
  • Variable assignment - Update variable values

Level 4:

  • Hash table lookup - Efficiently retrieve frequency values
  • Math operations - Calculate indexes using modulo

Level 5:

  • Array searching - Find first available element that satisfies condition
  • State update - Change object attributes to represent new state

The overall problem solving approach would be:

  1. Parse inputs
  2. Create objects to model problem
  3. Sort requests by arrival
  4. Loop through requests
    • Use conditional logic to check availability
    • Calculate assignment index
    • Search for first available server
    • Update object states to assign request
  5. Analyze updated state to find busiest server

Each concept builds on simpler ones, leading from basic iteration and conditionals to more complex state updates and data structure manipulations. Combining these units solves the overall problem.

Targeted Drills in Python

Here are some example Python coding drills for the key concepts:

  1. Parsing inputs
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# Read data from arrays into variables
arrival = [1, 2, 3] 
load = [5, 3, 2]
requests = list(zip(arrival, load))
  1. Sorting
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# Sort list of tuples by first element  
requests.sort(key=lambda x: x[0])
  1. Object instantiation
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class Server:
  def __init__(self, id):
    self.id = id
    self.available = True
    self.load = 0
  1. Looping
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for request in requests:
  # process request
  1. Conditionals
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if server.available:
  # assign request 
else:
  # find another server
  1. Hash table lookup
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frequency = {} 

frequency[item] += 1

These pieces could be integrated sequentially:

  1. Parse inputs
  2. Create Server and Request objects
  3. Sort Requests
  4. Loop through Requests
    • Use if/else logic on availability
    • Lookup/update frequencies
    • Update server loads
  5. Analyze frequencies to find busiest

Building up each concept into modular pieces enables flexibly assembling them to solve the full problem.

Q&A

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