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Master-Level Question 1: Graph Optimization
Question:
Given a weighted graph, implement Dijkstra’s algorithm to determine the shortest distance from a source vertex to every other vertex. Explain the time complexity when a binary heap is used.
Expert Solution:
A priority queue can efficiently select the unvisited vertex with the smallest tentative distance. Initialize the source distance to zero and every other distance to infinity. Repeatedly remove the minimum-distance vertex and relax each outgoing edge.
Using an adjacency list and binary heap, the overall complexity is O((V + E) log V), where V represents vertices and E represents edges. This approach is suitable for graphs containing non-negative edge weights.
A programming assignment helper can also assist students in checking edge cases, explaining the algorithm, and improving implementation efficiency.
Master-Level Question 2: Dynamic Programming
Question:
Design an algorithm to solve the 0/1 Knapsack problem, where each item has a weight and value, and each item can either be selected once or rejected.
Expert Solution:
Define dp[i][w] as the maximum value obtainable using the first i items with a capacity of w. For each item, compare two possibilities: excluding it or including it when its weight fits within the capacity.
The recurrence is:
dp[i][w] = max(dp[i-1][w], value[i] + dp[i-1][w-weight[i]])
The resulting dynamic-programming solution has O(nW) time complexity and O(nW) space complexity, where n is the number of items and W is the knapsack capacity. Space can be optimized to O(W) by maintaining a one-dimensional array.
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