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

Which statement about a recursive function without a reachable base case is correct?

  1. AIt keeps calling itself until the call stack overflows
  2. BIt returns 0 automatically
  3. CIt runs in O(1) time
  4. DThe compiler always turns it into a loop that ends safely
2.Algorithms & complexityHard

What did the Cook–Levin theorem establish?

  1. AThe halting problem is undecidable
  2. BP is not equal to NP
  3. CEvery NP problem can be solved in exponential time
  4. DThe Boolean satisfiability problem (SAT) is NP-complete
3.Data structuresMedium

In a circular queue stored in an array of capacity n, how is the next position after index i computed?

  1. A2i + 1
  2. Bi + 1
  3. C(i + 1) mod n
  4. D(i − 1) mod n
4.Data structuresHard

What is the main advantage of a skip list over a sorted linked list?

  1. AIt stores keys without order
  2. BExpected O(log n) search by using several levels of forward pointers
  3. CIt uses no extra memory
  4. DGuaranteed O(1) search because every node stores a pointer to every other node
5.Graph theoryMedium

A connected graph has 6 vertices and 10 edges. How many edges k must be removed to leave a spanning tree?

|V| = 6|E| = 10
k =
6.Graph theoryHard
1413148136515ABCDEF

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 · It keeps calling itself until the call stack overflows

Why this is the answer

Without a base case the recursion never stops; in practice memory for stack frames runs out and the program crashes.

Answer

A · It keeps calling itself until the call stack overflows
2.Algorithms & complexity

Answer: D · The Boolean satisfiability problem (SAT) is NP-complete

Why this is the answer

Cook and Levin showed independently that every problem in NP reduces to SAT in polynomial time, so SAT was the first known NP-complete problem.

Answer

D · The Boolean satisfiability problem (SAT) is NP-complete
3.Data structures

Answer: C · (i + 1) mod n

Why this is the answer

The modulo operation wraps index n − 1 back to 0, so the array is reused without shifting elements.

Answer

C · (i + 1) mod n
4.Data structures

Answer: B · Expected O(log n) search by using several levels of forward pointers

Why this is the answer

Randomised levels allow searches to skip many nodes, giving expected O(log n) search, insert and delete.

Answer

B · Expected O(log n) search by using several levels of forward pointers
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

6 − 1 = 5

Edges to remove

k = 10 − 5 = 5

Answer

k = 5
6.Graph theory

Answer: d(F) = 20

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: 14, C: 13
C (13): E: 26
B (14): D: 22, E: 19
E (19): F: 20
F (20)
D (22)

Shortest path

A → B → E → F
d(F) = 20

Answer

d(F) = 20

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