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

The time complexity of naive recursive Fibonacci (fib(n) = fib(n−1) + fib(n−2)) is:

  1. AConstant
  2. BLinear
  3. CLogarithmic
  4. DExponential
2.Algorithms & complexityHard

What is the time complexity of building a binary heap from n unsorted elements with the bottom-up method?

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

In a binary heap stored in an array starting at index 0, where is the parent of the node at index i?

  1. AAt index i − 1
  2. BAt index ⌊(i − 1) / 2⌋
  3. CAt index 2i + 1
  4. DAt index ⌊i / 2⌋ + 1
4.Data structuresHard

How can you detect whether a singly linked list contains a cycle using O(1) extra memory?

  1. AMove one pointer by one step and another by two steps and check whether they meet
  2. BSort the list and look for duplicates
  3. CCount the nodes until the count exceeds the list length field
  4. DStore every visited node in a hash set
5.Graph theoryMedium

A connected graph has 6 vertices and 10 edges. How many edges k must be removed to leave a spanning tree?

|V| = 6|E| = 10
k =
6.Graph theoryHard
161658121411ABCDEF

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

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

3Solutions

1.Algorithms & complexity

Answer: D · Exponential

Why this is the answer

The call tree grows roughly like φⁿ (about 1.618ⁿ). Memoisation reduces it to O(n).

Answer

D · Exponential
2.Algorithms & complexity

Answer: C · O(n)

Why this is the answer

Summing sift-down costs over the levels gives a convergent series, so heapify runs in linear time.

Answer

C · O(n)
3.Data structures

Answer: B · At index ⌊(i − 1) / 2⌋

Why this is the answer

Children of i are at 2i + 1 and 2i + 2, so the parent of i is ⌊(i − 1) / 2⌋; with 1-based indexing it is ⌊i / 2⌋.

Answer

B · At index ⌊(i − 1) / 2⌋
4.Data structures

Answer: A · Move one pointer by one step and another by two steps and check whether they meet

Why this is the answer

In Floyd's algorithm, if there is a cycle the fast pointer eventually catches up with the slow one; otherwise the fast pointer reaches null.

Answer

A · Move one pointer by one step and another by two steps and check whether they meet
5.Graph theory

Answer: k = 5

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

6 − 1 = 5

Edges to remove

k = 10 − 5 = 5

Answer

k = 5
6.Graph theory

Answer: W = 32

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

AC (1): take
CE (5): take
BD (6): take
DE (8): take
BE (11): skip, closes a cycle
DF (12): take

Total weight

W = 1 + 5 + 6 + 8 + 12 = 32

Answer

W = 32

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