Flipkart Challenge Inventory Management

Logical Breakdown

• [ ] Subproblem Identification:
• [ ] Optimal Substructure:
• [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
• [ ] Optimization: (Matrix Exponentiation if 'A' is huge)

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Mathematical Rigor

State Definition

Let \(dp[i]\) be the state for the \(i\)-th subproblem.

Recurrence Relation

\[ dp[i] = ... \]

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Visualization

graph TD A["Current State"] --> B["Previous State 1"] A --> C["Previous State 2"]

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Complexity Analysis

Approach	Time Complexity	Space Complexity
Iterative DP	\(O(N)\)	\(O(1)\)

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Code Reference

package com.dsa.heaps;

import java.util.*;

/* * ============================================================================================ * PROBLEM: Flipkart Inventory Management (Maximize Profit with Deadlines) * ============================================================================================ * * --------------------- * 1. Logical Breakdown & Mathematical Formulation * --------------------- * Each item \(i\) has an expiration time \(A[i]\) and profit \(B[i]\). We can buy 1 item per * unit of time. * * Step 1: The Greedy Strategy * - If we sort items by expiration time, we can decide which items to "keep" in our * schedule as we progress through time. * - At any time \(t\), if we have already selected \(t\) items and we encounter a new item * with expiration \(A[i]\), we have two cases: * 1. If \(A[i] >\) current number of selected items: We can safely buy this item. * 2. If \(A[i] \le\) current number: We can only buy this item if its profit \(B[i]\) * is greater than the least profitable item we've already picked. * * Step 2: Min-Heap Implementation * 1. Pair \(A[i]\) and \(B[i]\) and sort all items by expiration time (\(A[i]\)) ascending. * 2. Initialize a Min-Heap to store profits of selected items. * 3. For each item: * - If \(A[i] >\) Heap.size(): * - Add profit \(B[i]\) to Heap. * - Else if \(B[i] >\) Heap.peek(): * - Replace the least profitable item: Heap.poll() and Heap.push(B[i]). * 4. Sum all elements in the Heap and return (Total Profit % 10^9 + 7). * * --------------------- * 2. Complexity Analysis * --------------------- * Time Complexity: * O(N log N) * - Sorting the items: O(N log N). * - Processing N items with heap operations: O(N log N). * * Space Complexity: * O(N) to store the heap and the sorted pairs. * * --------------------- * 3. Base Case Verification * --------------------- * - All deadlines = 1: We can only pick 1 item (the one with max profit). * - Deadlines are large: We can pick all items. * ============================================================================================ / public class FlipkartChallengeInventoryManagement { public int solve(int A) { // TODO: Implement solution return 0; }

public int solve(int[] a3, int[] b3) {
    return 1;
}

}