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

Which sorting algorithm is stable?

  1. AHeap sort
  2. BMerge sort
  3. CQuicksort (typical in-place version)
  4. DSelection sort (typical version)
2.Algorithms & complexityHard

Why does any comparison-based search in a sorted array need Ω(log n) comparisons in the worst case?

  1. AEvery element must be compared at least once
  2. BComparisons take logarithmic time each
  3. CA decision tree with n possible outcomes must have depth at least log₂ n
  4. DBinary search is the only possible algorithm, and it needs log₂ n comparisons
3.Data structuresMedium

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

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

In a hash table using open addressing, why is deletion often done with a "tombstone" marker?

  1. AIt makes the table sorted
  2. BEmptying the slot could break the probe sequence for other keys
  3. CDeleted keys must be kept forever for security, so slots can never be reused
  4. DTombstones speed up hashing
5.Graph theoryMedium

A tree has 2 vertices of degree 3, 2 vertices of degree 4, 2 vertices of degree 5, 2 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

2 × deg 32 × deg 42 × deg 52 × deg 2
L =
6.Graph theoryHard
ABCDEF

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

M = adjacency matrixL = 2
N =

3Solutions

1.Algorithms & complexity

Answer: B · Merge sort

Why this is the answer

Merge sort is stable when merging takes from the left half first on ties. In-place quicksort, heap sort and selection sort are usually not stable.

Answer

B · Merge sort
2.Algorithms & complexity

Answer: C · A decision tree with n possible outcomes must have depth at least log₂ n

Why this is the answer

Each comparison at most halves the set of remaining possibilities, so distinguishing n positions needs at least log₂ n comparisons.

Answer

C · A decision tree with n possible outcomes must have depth at least log₂ n
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: B · Emptying the slot could break the probe sequence for other keys

Why this is the answer

If a deleted slot became empty, later searches might stop early and miss keys that were placed further along the probe chain.

Answer

B · Emptying the slot could break the probe sequence for other keys
5.Graph theory

Answer: L = 14

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.

Handshake lemma for a tree

L + 28 = 2(L + 8) − 2
L = 14

Answer

L = 14

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

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 1 1 0 0
C: 1 1 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)AA = 0·0 + 1·1 + 1·1 + 0·0 + 0·0 + 0·0 = 2

Answer

N = 2

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