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

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

AO(n log n)
BO(1)
CO(n)
DO(log n)

2.Algorithms & complexityHard

What is the time complexity of: for i = 1 to n: for (j = i; j ≤ n; j = j + i) do O(1) work?

AO(n log n)
BO(n √n)
CO(n)
DO(n²)

3.Data structuresMedium

How many nodes does a perfect binary tree of height h have, with height counted in edges?

Ah²
B2h + 1
C2^h
D2^h+1 − 1

4.Data structuresHard

A dynamic array doubles when full. Why do many implementations shrink it only when it is one quarter full, not half full?

ATo keep the array sorted
BTo avoid repeated resizing when pushes and pops alternate near the boundary
CBecause halving at one half would lose elements
DBecause the memory allocator can only release memory in quarters of a block

5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K₈ have? Circuits that differ only in the starting vertex or the direction count as the same.

K₈

H =

6.Graph theoryHard

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

|V| = 6|E| = 8

W =

3Solutions

1.Algorithms & complexity

Answer: D · O(log n)

Why this is the answer

The problem halves at each step with constant work per step, so there are about log₂ n steps.

Answer

D · O(log n)

2.Algorithms & complexity

Answer: A · O(n log n)

Why this is the answer

The total is n / 1 + n / 2 + ... + n / n = n × H_n, and the harmonic number H_n is Θ(log n).

Answer

A · O(n log n)

3.Data structures

Answer: D · 2^h+1 − 1

Why this is the answer

Summing 1 + 2 + 4 + ... + 2^h gives 2^h+1 − 1; for example, height 2 gives 7 nodes.

Answer

D · 2^h+1 − 1

4.Data structures

Answer: B · To avoid repeated resizing when pushes and pops alternate near the boundary

Why this is the answer

Shrinking at one half could trigger a resize on every operation in such a sequence; the gap between 1/4 and full keeps the amortised cost O(1).

Answer

B · To avoid repeated resizing when pushes and pops alternate near the boundary

5.Graph theory

Answer: H = 2520

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.

Count

H = (8 − 1)!/2 = 5040/2 = 2520

Answer

H = 2520

6.Graph theory

Answer: W = 52

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

DE (1): take

BE (4): take

BD (6): skip, closes a cycle

CE (14): take

BC (15): skip, closes a cycle

AC (16): take

DF (17): take

Total weight

W = 1 + 4 + 14 + 16 + 17 = 52

Answer

W = 52

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