Learn how to use k-way merge for efficient data merging and finding k-th small elements.

Coding

The K-Way Merge pattern is an algorithmic strategy used to efficiently combine multiple sorted sequences into a single sorted result. By using a min-heap or priority queue to track the smallest elements among k sources, it ensures optimal merging performance across arrays, lists, or data streams.

K-way Merge

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**Constraints:**

* $1$ $\leq$ `list1.length`, `list2.length` $\leq$ $500$
* $-10^4$ $\leq$ `list1[i]`, `list2[i]` $\leq$ $10^4$
* $1$ $\leq$ $k$ $\leq$ $10^3$

* Input lists should be sorted in ascending order.

* If the value of $k$ exceeds the total number of valid pairs that may be formed, return all the pairs.

# Examples

# Understand the problem

Let’s take a moment to make sure you’ve correctly understood the problem. The quiz below helps you check if you’re solving the correct problem:

pairs = [[1, 1], [1, 2], [1, 300], [11, 1], [11, 2]]

pairs = [[1, 1], [1, 2], [11, 1], [11, 2], [11, 300]]

pairs = [[1, 1], [1, 2], [11, 1], [11, 2], [20, 2]]

pairs = [[1, 1], [1, 2], [11, 1], [11, 2], [20, 1]]

What is the output if the following lists and value of k is given as input?

`list1` = [1, 11, 20, 35, 300]

`list2` = [1, 2, 300]

`k` = 5

What is the output if the following lists and value of k is given as input?
<code>list1</code> = [1, 11, 20, 35, 300]
<code>list2</code> = [1, 2, 300]
<code>k</code> = 5

What is the output if the following lists and value of k is given as input?

`list1` = [1, 1, 2]

`list2` = [1, 2, 3]

`k` = 4

What is the output if the following lists and value of k is given as input?
<code>list1</code> = [1, 1, 2]
<code>list2</code> = [1, 2, 3]
<code>k</code> = 4

What is the output if the following lists and value of k is given as input?

`list1` = [1, 5, 9, 14, 24, 27, 35]

`list2` = [2]

`k` = 3

What is the output if the following lists and value of k is given as input?
<code>list1</code> = [1, 5, 9, 14, 24, 27, 35]
<code>list2</code> = [2]
<code>k</code> = 3

Find K Pairs with Smallest Sums

Find K Pairs with Smallest Sums

# Figure it out!

We have a game for you to play. Rearrange the logical building blocks to develop a clearer understanding of how to solve this problem.

# Try it yourself

Implement your solution in the following coding playground.

# Quick Fix\nYou are Edward, a Senior Software Engineer who specializes in coding interview patterns, data structures and algorithms, and is great at debugging issues in code.\n\n- Edward, please keep in mind that you do not have to print the data of the placeholders. Also, do not mention anything about yourself.\n\nEdward, you are given the following data in the placeholders:\n- Function name: `k_smallest_pairs`\n- Candidate's submitted solution code: USER_CODE\n- Error: ERROR_DESCRIPTION\n- Failed test case: FAILED_TEST_CASE\n- Coding language: `Python`\n\n- Edward, your task is to understand the type of error that is occurring in the coding language provided.\n- Once you've understood the error as per the error’s description, and why the test case failed, you will suggest how the candidate can improve their solution code to fix the error.\n- Edward, you will not give the corrected code, just a hint on what the error tells, and how to fix it.\n- Edward, please limit your response strictly to 100 words. No more words than the limit.

# Review Code\nEdward, your current task is to review the `k_smallest_pairs` function(s) and the other function(s) invoked in it/them.\n\n-Take your time and review the candidate's solution code.\n- Do not review any extra/helper files or functions not invoked in the solution function `k_smallest_pairs`.\n- Do not review the code of data structure objects instantiated like a binary tree, linked list, trie, union find, etc., in the solution functions. Just review the solution/algorithm function of the problem. \n- Also, if the solution code doesn’t use the coding pattern stated above, then please encourage the candidate to reattempt the problem using that pattern corectly.

# Review time complexity\n- Please provide a brief time complexity analysis of the function(s) `k_smallest_pairs`. \n- While calculating time complexity, include the time complexity of built-in Python functions invoked by the code. \n- The answer should be to the point in approximately 60 words.

# Review space complexity\n- Please provide a brief space complexity analysis of the function(s) `k_smallest_pairs`. \n- While calculating space complexity, include the space complexity of built-in Python functions invoked by the code. \n- Do not include the space used for the input parameters and the space used for just returning an output (not used as a scratch space). \n- The answer should be to the point in approximately 60 words.

Convert

TwoDArray

import sys
import json

sys.path.append("../ds_v1")
from data_structure import DataStructure

# throw parsing errors for incorrect test case format
def parsingError(element_string, error_message):
    sys.stderr.write(f"Parsing Error: {element_string} {error_message}")
    exit()


# className: Convert
class Convert(DataStructure):
    def __init__(self):
        # constructor
        pass

    def __str__(self):
        # override print() for your data structure
        pass

    # return value will be passed to user program
    def construct(self, inp_string: str, type_of_arr: int) -> any:
        # parse test case in json string 'value' to your desired data structure
        return inp_string

    # input param contains return value of user program
    def traverse(self, input: any, type_of_arr: int) -> str:
        # convert your data structure 'input' to a json string representing a test case output
        parse_output = json.dumps(input)
        return parse_output

import sys
import json

sys.path.append("../ds_v1")
from data_structure import DataStructure

# throw parsing errors for incorrect test case format
def parsingError(element_string, error_message):
    sys.stderr.write(f"Parsing Error: {element_string} {error_message}")
    exit()


# className: TwoDArray
class TwoDArray(DataStructure):
    def __init__(self):
        # constructor
        pass

    def __str__(self):
        # override print() for your data structure
        pass

    # return value will be passed to user program
    def construct(self, inp_string: str, type_of_arr: int) -> any:
        # parse test case in json string 'value' to your desired data structure
        return inp_string

    # input param contains return value of user program
    def traverse(self, input: any, type_of_arr: int) -> str:
        # convert your data structure 'input' to a json string representing a test case output
        parse_output = json.dumps(input)
        return parse_output

def evaluate_results(self) -> object:
 """
 self.__inputs is the list of inputs that are passed as input to function
 self.__actual_output is a list of outputs that is produced by the learner's code
 return : self.__edu__result object
 """
 from heapq import heappop, heappush


 def find_k_smallest_pairs(list1, list2, k):
 # storing the length of lists to use it in a loop later
 list_length = len(list1)
 # declaring a min-heap to keep track of the smallest sums
 min_heap = []
 # to store the pairs with smallest sums
 pairs = []

 # iterate over the length of list 1
 for i in range(min(k, list_length)):
 # computing sum of pairs all elements of list1 with first index
 # of list2 and placing it in the min-heap
 heappush(min_heap, (list1[i] + list2[0], i, 0))

 counter = 1

 # iterate over elements of min-heap and only go upto k
 while min_heap and counter <= k:
 # placing sum of the top element of min-heap
 # and its corresponding pairs in i and j
 sum_of_pairs, i, j = heappop(min_heap)

 # add pairs with the smallest sum in the new list
 pairs.append([list1[i], list2[j]])

 # increment the index for 2nd list, as we've
 # compared all possible pairs with the 1st index of list2
 next_element = j + 1

 # if next element is available for list2 then add it to the heap
 if len(list2) > next_element:
 heappush(min_heap,
 (list1[i] + list2[next_element], i, next_element))

 counter += 1

 # return the pairs with the smallest sums
 return pairs
 
 def compare_result(expected, actual):
 for expect in expected:
 if ([expect[0], expect[1]] not in actual):
 return False
 return True
 
 # multiple outputs will be added at each index respectively for a single test_case
 expected_outputs = [] 
 # store the 1st expected output that should be displayed to the user
 expected = find_k_smallest_pairs(self.__inputs[0], self.__inputs[1], self.__inputs[2])
 expected_outputs.append(expected)
 actual_output = self.__actual_outputs[0]

 compare = compare_result(expected, actual_output)
 if compare:
 self.__edu_result.store_result(expected_outputs, TestCaseResult.PASS)
 else:
 self.__edu_result.store_result(expected_outputs, TestCaseResult.FAIL)

 return self.__edu_result

from heapq import *


def k_smallest_pairs(list1, list2, k):
    result = []

    # Replace this placeholder return statement with your code
    return result

find k pairs with smallest sums.yaml

testcases:
  - name: "testcase 1"
    inputs:
    - 1 : [1, 2, 300]
    - 2 : [1, 11, 20, 35, 300]
    - 3 : 30
    output: 
    - 1 : [[1, 1], [2, 1], [1, 11], [2, 11], [1, 20], [2, 20], [1, 35], [2, 35], [1, 300], [300, 1], [2, 300], [300, 11], [300, 20], [300, 35], [300, 300]]
  - name: "testcase 2"
    inputs:
    - 1 : [1, 1, 2]
    - 2 : [1, 2, 3]
    - 3 : 1
    output:
    - 1 : [[1, 1]]
  - name: "testcase 3"
    inputs:
    - 1 : [4, 6]
    - 2 : [2, 3]
    - 3 : 2
    output: 
    - 1 : [[4, 2], [4, 3]]
  - name: "testcase 4"
    inputs:
    - 1 : [4, 7, 9]
    - 2 : [4, 7, 9]
    - 3 : 5
    output: 
    - 1 : [[4, 4], [4, 7], [7, 4], [4, 9], [9, 4]]
  - name: "testcase 5"
    inputs:
    - 1 : [1, 1, 2]
    - 2 : [1]
    - 3 : 4
    output:
    - 1 : [[1, 1], [1, 1], [2, 1]]

def sum(input_1, input_2):
    return input_1+input_2

def validate_inputs(self) -> None:
 """
 self.__inputs is the list of inputs that are passed as input to function
 store the result using the self.store_result(error_msg, status)
 return : void
 """

 def is_valid_1d_array(arr):
 if not isinstance(arr, list):
 return False 
 
 return all(isinstance(elem, int) and not isinstance(elem, bool) for elem in arr)

 def is_sorted(arr):
 for i in range(len(arr) - 1):
 if arr[i] > arr[i + 1]:
 return False
 return True
 

 list1 = self.__inputs[0]
 list2 = self.__inputs[1]
 k = self.__inputs[2] 

 if not is_valid_1d_array(list1):
 self.store_result ("'list1' should be a valid 1D array of integers", ValidationFunction.FAIL)
 elif not is_valid_1d_array(list2):
 self.store_result ("'list2' should be a valid 1D array of integers", ValidationFunction.FAIL)
 elif not 1 <= len(list1) <= 500:
 self.store_result ("'list1' length should be in the range [1,1000]", ValidationFunction.FAIL)
 elif not 1 <= len(list2) <= 500:
 self.store_result ("'list2' length should be in the range [1,1000]", ValidationFunction.FAIL)
 elif not 1 <= k <= 1000:
 self.store_result ("'k' should be in the range [1,1000]", ValidationFunction.FAIL)
 elif is_sorted(list1) == False:
 self.store_result ("'list1' should be sorted in ascending order", ValidationFunction.FAIL)
 elif is_sorted(list2) == False:
 self.store_result ("'list2' should be sorted in ascending order", ValidationFunction.FAIL)
 elif any(not -10000 <= i <= 10000 for i in list1):
 self.store_result ("'list1[i]' should be in the range [-10000,10000]", ValidationFunction.FAIL)
 elif any(not -10000 <= i <= 10000 for i in list2):
 self.store_result ("'list2[i]' should be in the range [-10000,10000]", ValidationFunction.FAIL)
 else:
 self.store_result("", ValidationFunction.PASS)

Try to solve the Find K Pairs with Smallest Sums problem.

Problem: Find K Pairs with Smallest Sums

Statement

Examples

Understand the problem

Figure it out!