Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

New problemsBack to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

Which algorithm finds the shortest paths from one source in a graph with non-negative edge weights?

  1. ADijkstra's algorithm
  2. BDepth-first search
  3. CBellman–Ford is required in all cases
  4. DKruskal's algorithm
2.Algorithms & complexityHard

For which knapsack variant does the greedy rule "take items in order of value per weight" always give an optimal answer?

  1. ANeither variant
  2. BThe fractional knapsack, where items may be split
  3. CThe 0/1 knapsack, where items must always be taken whole
  4. DBoth variants
3.Data structuresMedium

A node in a binary search tree has a right subtree. Where is its in-order successor?

  1. AThe leftmost node of its right subtree
  2. BThe rightmost node of its left subtree
  3. CIts parent, always
  4. DIts right child, always
4.Data structuresHard

Which pair of traversals always determines a unique binary tree with distinct keys?

  1. ALevel-order alone
  2. BPreorder and postorder
  3. CPreorder alone
  4. DPreorder and inorder
5.Graph theoryMedium

A tree has 2 vertices of degree 3, 2 vertices of degree 4, 2 vertices of degree 5, 2 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

2 × deg 32 × deg 42 × deg 52 × deg 2
L =
6.Graph theoryHard
31011374136126ABCDEFG

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 3; A itself is round 1.

start: Around 3
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F
  7. 7G

3Solutions

1.Algorithms & complexity

Answer: A · Dijkstra's algorithm

Why this is the answer

Dijkstra's algorithm works with non-negative weights; Bellman–Ford handles negative weights; Kruskal builds minimum spanning trees.

Answer

A · Dijkstra's algorithm
2.Algorithms & complexity

Answer: B · The fractional knapsack, where items may be split

Why this is the answer

When items can be split, the best ratio first is optimal; in 0/1 knapsack a high-ratio item can block a better combination, so dynamic programming is needed.

Answer

B · The fractional knapsack, where items may be split
3.Data structures

Answer: A · The leftmost node of its right subtree

Why this is the answer

All larger keys close to the node are in its right subtree, and the smallest of those is the leftmost node there.

Answer

A · The leftmost node of its right subtree
4.Data structures

Answer: D · Preorder and inorder

Why this is the answer

Preorder gives the root, and its position in the inorder sequence splits the left and right subtrees; preorder with postorder is ambiguous when a node has one child.

Answer

D · Preorder and inorder
5.Graph theory

Answer: L = 14

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Handshake lemma for a tree

L + 28 = 2(L + 8) − 2
L = 14

Answer

L = 14

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: C

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Dijkstra round by round

1. A (0): B: 3, C: 10
2. B (3): C: 4, D: 16
3. C (4): E: 11, G: 16
4. E (11): D: 15, F: 17
5. D (15)
6. G (16)
7. F (17)

round 3

C

Answer

C

ExamProblems · examproblems.com/courses/data-structures-and-algorithms