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(K_n)| = n(n − 1)/2, |E(K_m,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 K_n: (n − 1)!/2
Dijkstra update: d(v) = min(d(v), d(u) + w(u, v))
Walks of length L: (Mᴸ)_ij

Method

Write down the degree of every vertex; the degrees add up to twice the number of edges.
For Euler questions, check that the graph is connected and count the vertices with odd degree.
For trees, use n − 1 edges and the handshake lemma.
For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
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

Why does dynamic programming give no real speed-up for merge sort?

AIts subproblems do not overlap, so there is nothing to reuse
BMerge sort has no recursive structure
CMerge sort is already O(n)
DDynamic programming only works on graph problems, not on arrays

2.Algorithms & complexityHard

Which bound is correct for log(n!)?

AΘ(n)
BΘ(n log n)
CΘ(n²)
DΘ(log n)

3.Data structuresMedium

An adjacency matrix for a graph with V vertices uses how much memory?

AO(V²)
BO(log V)
CO(E)
DO(V + E)

4.Data structuresHard

What guarantee do the red-black tree rules give about the height of a tree with n keys?

AThe height is at most log₂ n + 1
BThe height is at most 2 log₂(n + 1)
CThe height is at most n / 2
DThe height is exactly log₂ n

5.Graph theoryMedium

A connected graph has 15 vertices and 23 edges. How many edges k must be removed to leave a spanning tree?

|V| = 15|E| = 23

k =

6.Graph theoryHard

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F

d(F) =

3Solutions

1.Algorithms & complexity

Answer: A · Its subproblems do not overlap, so there is nothing to reuse

Why this is the answer

Merge sort's halves are disjoint and each is solved once; DP helps only when subproblems recur, as in Fibonacci.

Answer

A · Its subproblems do not overlap, so there is nothing to reuse

2.Algorithms & complexity

Answer: B · Θ(n log n)

Why this is the answer

log(n!) ≤ n log n, and the largest n / 2 factors give log(n!) ≥ (n / 2) log(n / 2); this is why comparison sorting needs Ω(n log n).

Answer

B · Θ(n log n)

3.Data structures

Answer: A · O(V²)

Why this is the answer

An adjacency matrix stores V × V entries, efficient for dense graphs; adjacency lists use O(V + E), better for sparse graphs.

Answer

A · O(V²)

4.Data structures

Answer: B · The height is at most 2 log₂(n + 1)

Why this is the answer

Every path has the same number of black nodes and red nodes cannot be adjacent, so the longest path is at most twice the shortest, giving height O(log n).

Answer

B · The height is at most 2 log₂(n + 1)

5.Graph theory

Answer: k = 9

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; K_n 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

15 − 1 = 14

Edges to remove

k = 23 − 14 = 9

Answer

k = 9

6.Graph theory

Answer: d(F) = 24

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

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): B: 4, C: 11

B (4): C: 5, D: 12, G: 19

C (5): E: 14

D (12): F: 24, G: 16

E (14)

G (16)

F (24)

Shortest path

A → B → D → F

d(F) = 24

Answer

d(F) = 24

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