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

What does the recurrence T(n) = 2T(n − 1) + O(1) solve to?

  1. AO(2ⁿ)
  2. BO(n log n)
  3. CO(n²)
  4. DO(n)
2.Algorithms & complexityHard

Which of these satisfiability problems is known to be solvable in polynomial time?

  1. ACircuit satisfiability
  2. B2-SAT, where each clause has at most two literals
  3. C3-SAT, where each clause has exactly three literals
  4. DGeneral SAT on formulas in conjunctive normal form
3.Data structuresMedium

The keys 50, 30, 70, 20, 40 are inserted in this order into an empty binary search tree. What is the left child of 30?

  1. A50
  2. B70
  3. C20
  4. D40
4.Data structuresHard

Deleting a node with two children from a binary search tree is usually done by:

  1. ASwapping it with the root
  2. BReplacing it with its in-order successor (or predecessor) and deleting that node
  3. CRemoving the whole subtree
  4. DRebuilding the subtree below it from scratch and reinserting every key it contained
5.Graph theoryMedium

A connected graph has 17 vertices and 24 edges. How many edges k must be removed to leave a spanning tree?

|V| = 17|E| = 24
k =
6.Graph theoryHard
8421271388ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 4; A itself is round 1.

start: Around 4
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F

3Solutions

1.Algorithms & complexity

Answer: A · O(2ⁿ)

Why this is the answer

The number of calls doubles with each level and there are n levels, so the total is about 2ⁿ, as in the Towers of Hanoi.

Answer

A · O(2ⁿ)
2.Algorithms & complexity

Answer: B · 2-SAT, where each clause has at most two literals

Why this is the answer

2-SAT is in P (linear time via strongly connected components), while 3-SAT is NP-complete.

Answer

B · 2-SAT, where each clause has at most two literals
3.Data structures

Answer: C · 20

Why this is the answer

20 is smaller than 50 and smaller than 30, so it becomes the left child of 30; 40 becomes its right child.

Answer

C · 20
4.Data structures

Answer: B · Replacing it with its in-order successor (or predecessor) and deleting that node

Why this is the answer

The in-order successor has at most one child, so it can be removed easily after copying its key into the node being deleted.

Answer

B · Replacing it with its in-order successor (or predecessor) and deleting that node
5.Graph theory

Answer: k = 8

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

17 − 1 = 16

Edges to remove

k = 24 − 16 = 8

Answer

k = 8
6.Graph theory

Answer: E

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.

Dijkstra round by round

1. A (0): B: 8, C: 4
2. C (4): B: 6, E: 11
3. B (6): D: 18
4. E (11): F: 19
5. D (18)
6. F (19)

round 4

E

Answer

E

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