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

Dynamic programming is useful when a problem has:

  1. AOverlapping subproblems and optimal substructure
  2. BOnly random inputs
  3. CA single loop
  4. DNo subproblems at all, so it can be solved in one step
2.Algorithms & complexityHard

The halting problem shows that:

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

A queue is implemented with two stacks: push onto an "in" stack, and pop from an "out" stack, refilling it from "in" when empty. What is the amortised cost per operation?

  1. AO(1)
  2. BO(n²)
  3. CO(log n)
  4. DO(n)
4.Data structuresHard

Which structure lets you compute the maximum of every window of size k in an array in O(n) total?

  1. AA sorted array updated by shifting
  2. BA deque holding indices in decreasing order of value
  3. CA singly linked list of the window's elements
  4. DA min-heap rebuilt for every window
5.Graph theoryMedium

A tree has 2 vertices of degree 3, 3 vertices of degree 4, 4 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 33 × deg 44 × deg 52 × deg 2
L =
6.Graph theoryHard
616132109517ABCDEF

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 6|E| = 8
W =

3Solutions

1.Algorithms & complexity

Answer: A · Overlapping subproblems and optimal substructure

Why this is the answer

Storing results of overlapping subproblems (memoisation or tabulation) avoids repeated work, turning exponential algorithms into polynomial ones.

Answer

A · Overlapping subproblems and optimal substructure
2.Algorithms & complexity

Answer: B · 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

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

Answer: A · O(1)

Why this is the answer

Each element is pushed and popped at most twice in total, so although one refill can take O(n), the average over a sequence is O(1).

Answer

A · O(1)
4.Data structures

Answer: B · A deque holding indices in decreasing order of value

Why this is the answer

The front of the deque is always the window's maximum; each index enters and leaves the deque once, so the work is linear.

Answer

B · A deque holding indices in decreasing order of value
5.Graph theory

Answer: L = 22

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 + 42 = 2(L + 11) − 2
L = 22

Answer

L = 22

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

6.Graph theory

Answer: W = 35

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.

Kruskal’s algorithm

CE (2): take
EF (5): take
AB (6): take
DF (9): take
DE (10): skip, closes a cycle
BC (13): take

Total weight

W = 2 + 5 + 6 + 9 + 13 = 35

Answer

W = 35

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