Algorithms

Advanced Practice - Algorithms

Practice balanced trees, heaps, dynamic programming, greedy algorithms, Dijkstra shortest paths, and tries with challenging problems.

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Algorithms
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About This Algorithm Practice

Work through challenging advanced problems step by step

These advanced exercises cover shortest path algorithms, minimum spanning tree construction, advanced dynamic programming table-filling, and backtracking, each with a full solution.

The problems are untimed so you can concentrate on mastering the techniques.

Finish the set to get ready for the expert-level challenge exercises.

What You Will Practice at Advanced Level

Balancing Trees

Exercises walk you through AVL and red-black rotations so you can restore the O(log n) height after inserts and deletes.

Heap Operations

You will practice building binary heaps and running O(log n) insert and extract-min, then apply them as priority queues.

Dynamic Programming Design

Guided problems help you find optimal substructure, write recurrences, and convert them into memoized or tabulated solutions.

Greedy, Dijkstra, and Tries

You will practice proving greedy choices, tracing Dijkstra at O((V+E) log V) with a heap, and building tries for prefix search.

Sample Questions

A few real questions from this test, with answers and explanations. Take the full test above for the complete set.

Dijkstra's algorithm fails to produce correct results when:

Answer: The graph contains negative edge weights

Dijkstra's algorithm assumes that once a node's shortest distance is finalized, it cannot be improved. Negative edge weights can violate this assumption, causing incorrect results. Use Bellman-Ford for graphs with negative weights.

The 0/1 Knapsack problem has a time complexity of:

Answer: O(n * W) where W is the capacity

The dynamic programming solution builds a table of size n x W, where n is the number of items and W is the knapsack capacity. Each cell is computed in O(1), giving O(n * W) total. Note this is pseudo-polynomial since W is not bounded by n.

What is the time complexity of the Floyd-Warshall algorithm?

Answer: O(V^3)

Floyd-Warshall uses three nested loops over all V vertices, giving O(V^3) time complexity. It computes shortest paths between all pairs of vertices using dynamic programming.

What is the lower bound for comparison-based sorting algorithms?

Answer: O(n log n)

Any comparison-based sorting algorithm must make at least O(n log n) comparisons in the worst case. This is proven by the decision tree model: with n! possible orderings, the tree needs height log(n!) = O(n log n).

Amortized analysis is used to:

Answer: Find the average cost per operation over a sequence of operations

Amortized analysis determines the average performance of each operation in a worst-case sequence. For example, a dynamic array's append is O(1) amortized even though individual resizes cost O(n), because resizes happen infrequently.

Frequently Asked Questions

Find answers to common questions about this assessment

Insert nodes one at a time and check each ancestor's balance factor. When a subtree becomes unbalanced, identify the case as left-left, right-right, left-right, or right-left, then apply the matching single or double rotation. Drawing each rotation on paper for small trees makes the pattern reliable before you code it.

Start by writing a plain recursive solution, then look for repeated subproblems. Add memoization to cache results, which often turns exponential time into polynomial. Later convert to bottom-up tabulation. Practicing classics like the knapsack, longest common subsequence, and coin change teaches you to spot optimal substructure quickly.

Use a small weighted graph with non-negative edges. Keep a priority queue of tentative distances, repeatedly extract the closest node, and relax its outgoing edges. Track how distances shrink until each node is finalized. With a binary heap this runs in O((V+E) log V), which hands-on tracing helps you internalize.

Greedy fails when a locally best choice blocks a better overall solution. For example, greedy coin change gives wrong answers for some coin systems, and it cannot solve the general knapsack. Practice by testing your greedy idea against small counterexamples; if one breaks it, reach for dynamic programming instead.

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