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

A function runs one loop of n iterations and then, separately, another loop of n iterations. What is its time complexity?

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

Why can Dijkstra's algorithm give wrong results when some edge weights are negative?

  1. AIt only works on undirected graphs
  2. BA vertex is finalised when extracted, but a later negative edge could still give it a shorter path
  3. CIt goes into an infinite loop on every negative edge
  4. DNegative numbers cannot be stored in a priority queue
3.Data structuresMedium

What is the value of the postfix expression 3 4 + 2 * ?

  1. A11
  2. B14
  3. C10
  4. D24
4.Data structuresHard

Under simple uniform hashing, what is the expected time of a search in a hash table with chaining, n keys and m buckets?

  1. AO(n)
  2. BO(m)
  3. CO(log n)
  4. DO(1 + α), where α = n / m
5.Graph theoryMedium

A connected graph has 16 vertices and 20 edges. How many edges k must be removed to leave a spanning tree?

|V| = 16|E| = 20
k =
6.Graph theoryHard
8719146143ABCDEF

Prim’s algorithm builds a minimum spanning tree starting from vertex B. Click the edge it adds as edge number 5.

start: Bk = 5
  1. 1edge AC
  2. 2edge BC
  3. 3edge BD
  4. 4edge CE
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge BE

3Solutions

1.Algorithms & complexity

Answer: D · O(n)

Why this is the answer

n + n = 2n, and constant factors are dropped, so the function is O(n).

Answer

D · O(n)
2.Algorithms & complexity

Answer: B · A vertex is finalised when extracted, but a later negative edge could still give it a shorter path

Why this is the answer

The greedy step relies on non-negative weights so that extracted distances are final; Bellman–Ford is used when weights may be negative.

Answer

B · A vertex is finalised when extracted, but a later negative edge could still give it a shorter path
3.Data structures

Answer: B · 14

Why this is the answer

Push 3 and 4, + gives 7; push 2, * gives 7 × 2 = 14.

Answer

B · 14
4.Data structures

Answer: D · O(1 + α), where α = n / m

Why this is the answer

Computing the hash costs O(1) and the expected chain length is α, so keeping α bounded by resizing gives O(1) expected time.

Answer

D · O(1 + α), where α = n / m
5.Graph theory

Answer: k = 5

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

16 − 1 = 15

Edges to remove

k = 20 − 15 = 5

Answer

k = 5
6.Graph theory

Answer: edge AC

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.

Prim’s algorithm

1. BE (3)
2. CE (1)
3. DE (4)
4. DF (6)
5. AC (8)

edge number 5

AC

Answer

edge AC

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