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(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

The time complexity of naive recursive Fibonacci (fib(n) = fib(n−1) + fib(n−2)) is:

  1. ALinear
  2. BExponential
  3. CLogarithmic
  4. DConstant
2.Algorithms & complexityHard

To show that a new problem X is NP-hard, what must you do?

  1. AReduce X to a known NP-hard problem in polynomial time
  2. BShow that X has no polynomial algorithm for small inputs
  3. CShow that X can be verified in polynomial time
  4. DReduce a known NP-hard problem to X in polynomial time
3.Data structuresMedium

Why can a binary heap be stored in an array without any pointers?

  1. AIt is a complete binary tree, so positions follow simple index formulas
  2. BAll keys in a heap are equal
  3. CHeaps never contain more than 100 elements
  4. DArrays automatically keep the keys in sorted order, so no pointers are needed
4.Data structuresHard

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

  1. AThe height is at most log2 n + 1
  2. BThe height is exactly log2 n
  3. CThe height is at most n / 2
  4. DThe height is at most 2 log2(n + 1)
5.Graph theoryMedium
ABCDEFGEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 7|E| = 7
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
7461341151418ABCDEFG

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: B · Exponential

Why this is the answer

The call tree grows roughly like φⁿ (about 1.618ⁿ). Memoisation reduces it to O(n).

Answer

B · Exponential
2.Algorithms & complexity

Answer: D · Reduce a known NP-hard problem to X in polynomial time

Why this is the answer

If a known hard problem can be transformed into X efficiently, a fast algorithm for X would solve the known problem too.

Answer

D · Reduce a known NP-hard problem to X in polynomial time
3.Data structures

Answer: A · It is a complete binary tree, so positions follow simple index formulas

Why this is the answer

Completeness means there are no gaps, so the children of index i are always at 2i + 1 and 2i + 2.

Answer

A · It is a complete binary tree, so positions follow simple index formulas
4.Data structures

Answer: D · 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

D · The height is at most 2 log2(n + 1)
5.Graph theory

Answer: neither

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.

Degrees

deg(A) = 2, deg(B) = 4, deg(C) = 2, deg(D) = 3, deg(E) = 1, deg(F) = 1, deg(G) = 1

Odd vertices

D, E, F, G → 4
neither

Answer

neither
6.Graph theory

Answer: d(F) = 22

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: 7
B (7): C: 11, D: 13, G: 21
C (11): E: 24
D (13): E: 17, F: 24, G: 14
G (14)
E (17): F: 22
F (22)

Shortest path

A → B → D → E → F
d(F) = 22

Answer

d(F) = 22

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