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

A greedy algorithm:

AUses only recursion
BAlways gives the optimal solution
CMakes the locally best choice at each step
DTries every possible solution

2.Algorithms & complexityHard

The Held–Karp dynamic programming algorithm solves the travelling salesman problem exactly. What is its running time for n cities?

AO(2^n²)
BO(n² 2ⁿ)
CO(n!)
DO(n³)

3.Data structuresMedium

You have a reference to a node in the middle of a list. In which list can you remove it in O(1)?

AA singly linked list
BA dynamic array
CA doubly linked list
DA sorted array

4.Data structuresHard

Without union by rank or path compression, what is the worst-case cost of a single find in a union-find structure with n elements?

AO(1)
BO(α(n))
CO(n)
DO(log n)

5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K₆ have? Circuits that differ only in the starting vertex or the direction count as the same.

K₆

H =

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: C · Makes the locally best choice at each step

Why this is the answer

Greedy algorithms are fast and optimal for some problems (e.g. Dijkstra, Huffman coding) but not in general (e.g. coin change with some coin systems).

Answer

C · Makes the locally best choice at each step

2.Algorithms & complexity

Answer: B · O(n² 2ⁿ)

Why this is the answer

There are about n 2ⁿ states, each computed in O(n); this is exponential but far faster than trying all (n − 1)! tours.

Answer

B · O(n² 2ⁿ)

3.Data structures

Answer: C · A doubly linked list

Why this is the answer

A doubly linked node knows its predecessor, so you can relink prev and next directly; a singly linked list must search for the predecessor.

Answer

C · A doubly linked list

4.Data structures

Answer: C · O(n)

Why this is the answer

Naive unions can link trees into a long chain, so find may follow n − 1 parent pointers; the two heuristics bring it close to constant.

Answer

C · O(n)

5.Graph theory

Answer: H = 60

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 = (6 − 1)!/2 = 120/2 = 60

Answer

H = 60

6.Graph theory

Answer: d(F) = 21

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: 5, C: 8

B (5): D: 11, G: 16

C (8): E: 18, G: 13

D (11)

G (13)

E (18): F: 21

F (21)

Shortest path

A → C → E → F

d(F) = 21

Answer

d(F) = 21

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