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Key Facts: VCE Algorithmics (HESS) Exam
VCE Algorithmics (HESS) is a university-level VCE subject co-designed by VCAA, Monash, and UMelb. This 100-question practice bank covers data modelling, dynamic programming, graph algorithms, big-O complexity, P vs NP, computability, and mathematical proofs.
Sample VCE Algorithmics (HESS) Practice Questions
Try these sample questions to test your VCE Algorithmics (HESS) exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.
1A connected simple undirected graph G has 15 vertices. Five vertices have degree 4, four vertices have degree 3, and six vertices have degree 2. How many edges does G contain?
A.22 edges
B.44 edges
C.15 edges
D.30 edges
Explanation: According to the Handshaking Lemma, the sum of all vertex degrees in a graph equals twice the number of edges: sum(deg(v)) = 2 * |E|. Here, the sum of degrees is (5 * 4) + (4 * 3) + (6 * 2) = 20 + 12 + 12 = 44. Dividing by 2 yields |E| = 22.
2A connected planar graph G has 12 vertices and 8 faces (regions). How many edges does G have according to Euler's Planar Graph Formula?
A.20 edges
B.18 edges
C.16 edges
D.22 edges
Explanation: Euler's formula for connected planar graphs states that V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. Substituting V = 12 and F = 8 gives 12 - E + 8 = 2, which simplifies to 20 - E = 2, so E = 18.
3What is the total number of edges in a complete undirected graph K_8, and what is its chromatic number chi(K_8)?
A.56 edges and chromatic number 8
B.28 edges and chromatic number 4
C.28 edges and chromatic number 8
D.36 edges and chromatic number 7
Explanation: A complete graph K_n with n vertices has n * (n - 1) / 2 edges. For n = 8, the number of edges is 8 * 7 / 2 = 28. Because every vertex is adjacent to every other vertex, no two vertices can share the same color, making chi(K_8) = 8.
4A complete bipartite graph K_{m,n} has 14 vertices in total. Which partition sizes m and n maximize the number of edges, and what is this maximum edge count?
A.m = 6, n = 8 with 48 edges
B.m = 1, n = 13 with 13 edges
C.m = 7, n = 7 with 98 edges
D.m = 7, n = 7 with 49 edges
Explanation: In a complete bipartite graph K_{m,n}, the total number of edges is m * n with constraint m + n = 14. The product m * n is maximized when m and n are as close as possible, so m = 7 and n = 7, yielding 7 * 7 = 49 edges.
5Consider a sparse connected graph G with V = 10,000 vertices and E = 20,000 edges. Which storage representation uses less memory space, and what is the time complexity to check whether an edge exists between vertex u and vertex v in an unsorted adjacency list?
A.Adjacency list uses less space; edge lookup takes O(deg(u)) time
B.Adjacency matrix uses less space; edge lookup takes O(1) time
C.Adjacency list uses less space; edge lookup takes O(1) time
D.Adjacency matrix uses less space; edge lookup takes O(V) time
Explanation: An adjacency list requires O(V + E) space, which for sparse graphs (E << V^2) is vastly smaller than the O(V^2) space of an adjacency matrix. To check for edge (u, v) in an unsorted adjacency list, one must iterate through the list of neighbors for vertex u, taking O(deg(u)) time.
6Let A be the 4x4 adjacency matrix of an undirected graph G. The entry (A^3)_{u,v} at row u and column v evaluates to 5. What does this value 5 represent?
A.The shortest path distance between vertex u and vertex v
B.The exact number of distinct walks of length 3 between vertex u and vertex v
C.The number of simple paths of length at most 3 between vertex u and vertex v
D.The degree of vertex u plus the degree of vertex v
Explanation: By graph theory matrix multiplication properties, the (u, v) entry of the k-th power of an adjacency matrix A^k gives the exact number of walks of length k between vertex u and vertex v in graph G.
7In a 0-indexed array implementation of a Min-Heap, if a parent node is located at array index i = 5, at which indices are its left child, right child, and parent node located?
A.Left child: 10, Right child: 11, Parent: 2
B.Left child: 11, Right child: 12, Parent: 3
C.Left child: 11, Right child: 12, Parent: 2
D.Left child: 12, Right child: 13, Parent: 1
Explanation: In a 0-indexed binary heap array, for node at index i: Left child index = 2*i + 1 = 2(5) + 1 = 11; Right child index = 2*i + 2 = 2(5) + 2 = 12; Parent index = floor((i - 1)/2) = floor(4/2) = 2.
8A complete binary tree has N = 63 nodes. Assuming height is measured by the number of edges on the longest root-to-leaf path, what is the height of this tree?
A.6
B.31
C.62
D.5
Explanation: A full/perfect binary tree of height h (edge-based) contains 2^(h+1) - 1 nodes. Setting 2^(h+1) - 1 = 63 gives 2^(h+1) = 64 = 2^6, so h + 1 = 6 and height h = 5.
9Performing an in-order traversal on a valid Binary Search Tree (BST) containing distinct numeric keys yields which sequence of key values?
A.The keys in strictly sorted ascending order
B.The keys in level-order priority sequence
C.The keys in post-order destruction sequence
D.The keys arranged with the root element placed first
Explanation: By definition of a Binary Search Tree, for every node, all keys in the left subtree are smaller and all keys in the right subtree are larger. An in-order traversal visits (Left, Root, Right), which outputs keys in strictly sorted ascending order.
10A binary tree has preorder traversal [A, B, D, E, C, F] and inorder traversal [D, B, E, A, F, C]. What is the root of the left subtree of root A, and what is the postorder traversal of the tree?
A.Subtree root D; postorder traversal: D, B, E, F, C, A
B.Subtree root B; postorder traversal: D, E, B, F, C, A
C.Subtree root E; postorder traversal: E, D, B, C, F, A
D.Subtree root B; postorder traversal: A, C, F, E, B, D
Explanation: Preorder [A, ...] shows A is the main root. Inorder [D, B, E, A, F, C] splits into left subtree [D, B, E] and right subtree [F, C]. In preorder, the left subtree elements [B, D, E] start with B, making B the root of the left subtree. Recursively reconstructing gives postorder: [D, E, B, F, C, A].
About the VCE Algorithmics (HESS) Practice Questions
Verified exam format metadata for VCE Algorithmics (HESS) Units 3 & 4 is pending. The practice questions above remain available while official exam length, timing, passing score, fee, and administrator details are reviewed.