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

A greedy algorithm:

  1. AUses only recursion
  2. BAlways gives the optimal solution
  3. CTries every possible solution
  4. DMakes the locally best choice at each step
2.Algorithms & complexityHard

What is the difference between an undecidable problem and an intractable one?

  1. ABoth mean the problem has no algorithm at all
  2. BIntractable problems cannot be solved by any algorithm, while undecidable ones only need exponential time
  3. CAn undecidable problem has no algorithm for all inputs; an intractable one has one, but it is too slow
  4. DUndecidable problems are in NP, while intractable problems are in P
3.Data structuresMedium

What is the time complexity of searching for a key in a balanced binary search tree?

  1. AO(n)
  2. BO(1)
  3. CO(log n)
  4. DO(n log n)
4.Data structuresHard

Which property keeps every B-tree perfectly balanced?

  1. AAll leaves are at the same depth
  2. BEach node stores one key
  3. CThe keys are stored only in the root
  4. DEvery node has exactly two children
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
ABCDE

How many closed walks of length 2 start and end at C? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =

3Solutions

1.Algorithms & complexity

Answer: D · 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

D · Makes the locally best choice at each step
2.Algorithms & complexity

Answer: C · An undecidable problem has no algorithm for all inputs; an intractable one has one, but it is too slow

Why this is the answer

The halting problem has no algorithm at all; an NP-hard problem such as TSP can be solved by brute force, just not in known polynomial time.

Answer

C · An undecidable problem has no algorithm for all inputs; an intractable one has one, but it is too slow
3.Data structures

Answer: C · O(log n)

Why this is the answer

A balanced tree has height about log₂ n, so a search follows one path from root to leaf.

Answer

C · O(log n)
4.Data structures

Answer: A · All leaves are at the same depth

Why this is the answer

Nodes split upwards when full and the tree only gets taller when the root splits, so all leaves always stay on the same level.

Answer

A · All leaves are at the same depth
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; 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.

Count

H = (6 − 1)!/2 = 120/2 = 60

Answer

H = 60
6.Graph theory

Answer: N = 2

Degrees: the degree of a vertex is the number of edges that meet it. Every edge has two ends, so the degrees add up to twice the number of edges: Σ deg(v) = 2|E| (the handshake lemma). The number of vertices with odd degree is therefore even. Kn has n(n − 1)/2 edges, a k-regular graph on n vertices has nk/2, and a tree on n vertices has n − 1 edges.

Adjacency matrix

A: 0 1 0 0 0
B: 1 0 1 1 1
C: 0 1 0 0 1
D: 0 1 0 0 1
E: 0 1 1 1 0

Common neighbours

(M²)CC = 0·0 + 1·1 + 0·0 + 0·0 + 1·1 = 2

Answer

N = 2

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