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

In the activity selection problem, which greedy rule always gives a maximum number of non-overlapping activities?

  1. APick the activity with the fewest overlaps first, then repeat with the compatible ones
  2. BPick the activity that finishes earliest, then repeat with the compatible ones
  3. CPick the shortest activity first
  4. DPick the activity that starts earliest
2.Algorithms & complexityHard

What does the recurrence T(n) = 2T(n/2) + n log n solve to?

  1. AΘ(n log n)
  2. BΘ(n²)
  3. CΘ(n)
  4. DΘ(n log² n)
3.Data structuresMedium

A road network has 100,000 junctions, each connected to about 4 others. Which representation is most suitable?

  1. AA single sorted array of vertices
  2. BAdjacency list
  3. CA binary heap of edges
  4. DAdjacency matrix
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 sorts the children of each node by frequency
  3. CIt merges chains of single-child nodes into one edge labelled with a string
  4. DIt keeps only the first letter of each word
5.Graph theoryMedium

A connected graph has 18 vertices and 23 edges. How many edges k must be removed to leave a spanning tree?

|V| = 18|E| = 23
k =
6.Graph theoryHard
ABCDEF

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

M = adjacency matrixL = 2
N =

3Solutions

1.Algorithms & complexity

Answer: B · Pick the activity that finishes earliest, then repeat with the compatible ones

Why this is the answer

Choosing the earliest finishing time can be proven optimal by an exchange argument; the other rules have counterexamples.

Answer

B · Pick the activity that finishes earliest, then repeat with the compatible ones
2.Algorithms & complexity

Answer: D · Θ(n log² n)

Why this is the answer

Level k does about n log(n / 2ᵏ) work, and summing over log n levels gives Θ(n log² n), the extended second case of the master theorem.

Answer

D · Θ(n log² n)
3.Data structures

Answer: B · Adjacency list

Why this is the answer

A matrix would need 1010 cells, almost all empty; an adjacency list only stores the roughly 400,000 edge entries.

Answer

B · Adjacency list
4.Data structures

Answer: C · 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

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

Answer: k = 6

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

18 − 1 = 17

Edges to remove

k = 23 − 17 = 6

Answer

k = 6
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 1 0 0 0
B: 1 0 0 1 0 0
C: 1 0 0 0 1 0
D: 0 1 0 0 1 1
E: 0 0 1 1 0 1
F: 0 0 0 1 1 0

Common neighbours

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

Answer

N = 2

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