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.

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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

What does the recurrence T(n) = 2T(n − 1) + O(1) solve to?

  1. AO(n)
  2. BO(n log n)
  3. CO(n²)
  4. DO(2ⁿ)
2.Algorithms & complexityHard

Which algorithm finds a minimum spanning tree by repeatedly adding the cheapest edge that does not form a cycle?

  1. AFloyd–Warshall
  2. BKruskal's algorithm
  3. CPrim's algorithm starting from one node
  4. DDijkstra's algorithm
3.Data structuresMedium

What balance condition does an AVL tree maintain at every node?

  1. AEvery node is coloured red or black
  2. BThe heights of the left and right subtrees differ by at most 1
  3. CAll leaves are at exactly the same depth
  4. DBoth subtrees of every node contain exactly the same number of nodes
4.Data structuresHard

Which property keeps every B-tree perfectly balanced?

  1. AEvery node has exactly two children
  2. BEach node stores one key
  3. CThe keys are stored only in the root
  4. DAll leaves are at the same depth
5.Graph theoryMedium

A connected graph has 18 vertices and 20 edges. How many edges k must be removed to leave a spanning tree?

|V| = 18|E| = 20
k =
6.Graph theoryHard
42818121119209ABCDEFG

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 7|E| = 9
W =

3Solutions

1.Algorithms & complexity

Answer: D · O(2ⁿ)

Why this is the answer

The number of calls doubles with each level and there are n levels, so the total is about 2ⁿ, as in the Towers of Hanoi.

Answer

D · O(2ⁿ)
2.Algorithms & complexity

Answer: B · Kruskal's algorithm

Why this is the answer

Kruskal sorts edges by weight and adds them if they connect different components; Prim grows one tree from a start vertex.

Answer

B · Kruskal's algorithm
3.Data structures

Answer: B · The heights of the left and right subtrees differ by at most 1

Why this is the answer

The balance factor (height left − height right) must be −1, 0 or 1; rotations restore it after inserts and deletes.

Answer

B · The heights of the left and right subtrees differ by at most 1
4.Data structures

Answer: D · All leaves are at the same depth

Why this is the answer

Nodes split upwards when full and the tree only gets taller when the root splits, so all leaves always stay on the same level.

Answer

D · All leaves are at the same depth
5.Graph theory

Answer: k = 3

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.

Edges of a spanning tree

18 − 1 = 17

Edges to remove

k = 20 − 17 = 3

Answer

k = 3
6.Graph theory

Answer: W = 56

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.

Kruskal’s algorithm

AC (2): take
AB (4): take
BC (8): skip, closes a cycle
GD (9): take
DF (11): take
DE (12): take
BD (18): take

Total weight

W = 2 + 4 + 9 + 11 + 12 + 18 = 56

Answer

W = 56

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