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

With coins of 1, 3 and 4 and an amount of 6, how many coins does the greedy "largest coin first" method use, compared to the optimum?

  1. AGreedy uses 3 coins; the optimum is 3
  2. BGreedy uses 4 coins; the optimum is 3
  3. CGreedy uses 3 coins; the optimum is 2
  4. DGreedy uses 2 coins; the optimum is 2
2.Algorithms & complexityHard

The halting problem shows that:

  1. ALoops should be avoided
  2. BSome programs run faster on faster computers, so halting depends on hardware
  3. CNo algorithm can decide for every program and input whether it halts
  4. DEvery program eventually halts
3.Data structuresMedium

What is the worst-case height of a binary search tree with n nodes if keys are inserted in sorted order?

  1. An (it becomes a linked list)
  2. B1
  3. Clog₂ n
  4. D√n
4.Data structuresHard

How does a radix tree (compressed trie) improve on a plain trie?

  1. AIt stores each word in a hash table at the root instead of in separate nodes
  2. BIt merges chains of single-child nodes into one edge labelled with a string
  3. CIt sorts the children of each node by frequency
  4. DIt keeps only the first letter of each word
5.Graph theoryMedium

How many different Hamilton circuits H does the complete graph K8 have? Circuits that differ only in the starting vertex or the direction count as the same.

K8
H =
6.Graph theoryHard
ABCDEF

How many walks of length 2 go from A to C? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =

3Solutions

1.Algorithms & complexity

Answer: C · Greedy uses 3 coins; the optimum is 2

Why this is the answer

Greedy takes 4 + 1 + 1 = 3 coins, but 3 + 3 uses only 2, so greedy is not optimal for this coin system; dynamic programming is.

Answer

C · Greedy uses 3 coins; the optimum is 2
2.Algorithms & complexity

Answer: C · No algorithm can decide for every program and input whether it halts

Why this is the answer

Turing showed the halting problem is undecidable: assuming a decider exists leads to a contradiction with a program that does the opposite.

Answer

C · No algorithm can decide for every program and input whether it halts
3.Data structures

Answer: A · n (it becomes a linked list)

Why this is the answer

Sorted insertions create a degenerate tree with height n, making searches O(n). Balanced trees prevent this.

Answer

A · n (it becomes a linked list)
4.Data structures

Answer: B · It merges chains of single-child nodes into one edge labelled with a string

Why this is the answer

Collapsing paths with no branching reduces the number of nodes and the memory used while keeping prefix search.

Answer

B · It merges chains of single-child nodes into one edge labelled with a string
5.Graph theory

Answer: H = 2520

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 = (8 − 1)!/2 = 5040/2 = 2520

Answer

H = 2520
6.Graph theory

Answer: N = 1

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 0
B: 1 0 1 1 1 0
C: 0 1 0 0 1 0
D: 0 1 0 0 1 1
E: 0 1 1 1 0 1
F: 0 0 0 1 1 0

Common neighbours

(M²)AC = 0·0 + 1·1 + 0·0 + 0·0 + 0·1 + 0·0 = 1

Answer

N = 1

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