Evolutionary and classification methods for local labor markets delineation

Abstract

This paper proposes some new evolutionary and classification methods for the delineation of local labor markets (LLMs) in areas where there are a large number of small localities with little labor interaction. The evolutionary methods presented here, based on previous works of Flórez-Revuelta et al. (Int J Autom Comput 5:10–21, 2008a; PPSN X, LNCS 5199:1011–1020, 2008b) and Martínez-Bernabeu et al. (Expert Syst Appl 39:6754–6766, 2012), decrease their computational times (up to a 99 %) without deteriorating the quality and robustness of the solutions. Also, in this work we avoid geographical contiguity constraints because such restrictions might reduce the realism of the process. Another contribution of this paper is related to the location of new services—hospitals, schools, employment centers, etc.—taking into account the labor mobility patterns. In this context, we present a cluster partitioning of k-means procedure, which captures the common aspects of all the potential solutions of these evolutionary algorithms and allows us to rank the LLMs foci, understood as the main centers of activity of the markets. The performance of the algorithms is analyzed through a real commuting dataset of the region of Aragón (Spain).

Appendices

Appendix 1: Evolutionary algorithm proposed in the paper

The structure of the algorithm consists of the following steps:

• Step 0 (Initialization) Set n_p (size of the population), n₀ (individuals generated in each step), g (number of iterations to establish the completion step), size_min (minimum size of a configuration), n_min (minimum amount of flow between two BSUs to be considered adjacent) and γ (minimum number of adjacent BSUs).

• Step 1 (Determination of initial configurations) Determine the initial configurations using the algorithm of Appendix 2. These configurations are taken as BSUs.

• Step 2 (Initial population) Determine an initial population of n_p individuals in such a way that, at least one of them, verify (1)–(3).

• Step 3 (Genetic operators) Apply genetic operators (mutation, crossover) chosen at random until n₀ valid individuals have been produced.

• Step 4 (Truncation) Select the best n_p individuals of the generated population according to their fitness values.

• Step 5 (Completion) If the best individual of the current population has not changed during the last g iterations, go to step 6. Otherwise, go to step 2.

• Step 6 (Final local step) A local search procedure to improve valid individuals similar to that proposed in Flórez-Revuelta et al. (2008b, Section 4.2) is carried out. In this step we take the original BSUs (and not the configurations) as element of analysis. We study whether some change of the composition of original configurations may improve the fitness values of the solution. In our experience this step has a residual character and only improves marginally the fitness value of the solution at an additional small CPU time.

In our empirical example we have taken n_p = 100, n₀ = 1, g = 100, β₁ = 0.70, β₂ = 0.75, β₃ = 5000, β₄ = 1500; size_min = 1, 100 and 250. All of these values coincide with those used in FCM08 except β₃ and β₄ for which we have taken the values in Alonso et al. (2008). Finally, for the connectivity constraint, we have taken γ = 10 and n_min = 25.

Appendix 2: Determination of initial configurations

The determination of the initial configurations is made using the following algorithm:

• Step 0: Initiation

  Fix size_min > 0 the minimum size of a configuration. Put n⁽⁰⁾ = n, ${\text{C}}_{\text{i}}^{\left( 0\right)} = {\text{B}}_{\text{i}}$ for i = 1,…, n⁽⁰⁾, C ⁽⁰⁾ = B and s = 0.

• Step 1: Check the stop condition

  Calculate β_min = $\min_{{{\text{i}} = 1, \ldots ,n^{{ ( {\text{s)}}}} }} \{ {{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}^{{\left( {\text{s}} \right)}} }} }\}$. If β_min ≥ size_min, stop. Otherwise, go to step 2.

• Step 2: Determine the configurations to be merged

  Let ${\mathbf{C}}_{\upbeta }^{{\left( {\text{s}} \right)}} \subseteq {\mathbf{C}}^{{\left( {\text{s}} \right)}}$ where ${\mathbf{C}}_{\upbeta }^{{\left( {\text{s}} \right)}} = \{ {{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} \in {\mathbf{C}}^{{\left( {\text{s}} \right)}} {\text{ such that T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}^{{\left( {\text{s}} \right)}} }} < {\text{size}}_{\hbox{min} } } \}$.

  Determine ${\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} \in {\mathbf{C}}^{{\left( {\text{s}} \right)}}$ such that ${\text{t}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} }} = \hbox{min} \{ {{\text{t}}_{{{\text{C}}_{{_{\text{l}} }}^{{\left( {\text{s}} \right)}} , {\text{C}}_{{_{\text{l'}} }}^{{\left( {\text{s}} \right)}} }} :{\text{C}}_{\ell }^{{\left( {\text{s}} \right)}} {\text{ or C}}_{\ell '}^{{\left( {\text{s}} \right)}} \, \in {\mathbf{C}}_{\upbeta }^{{\left( {\text{s}} \right)}} {\text{ and }}\ell \ne \ell {\prime } \, } \}$. When several ${\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} \in {\mathbf{C}}^{{\left( {\text{s}} \right)}}$ verify this condition, we choose the pair that maximizes the flow $\text{T}_{{C_{i}^{\left( s \right)} ,C_{j}^{\left( s \right)} }} + \text{T}_{{C_{j}^{\left( s \right)} ,C_{i}^{\left( s \right)} }}$, and, if possible with $\text{C}_{i}^{\left( s \right)} ,\text{C}_{j}^{\left( s \right)} \in {\mathbf{C}}_{\upbeta }^{\left( s \right)}$.

• Step 3: Merge the selected configurations

  Built ${\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} = {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} \cup {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}}$ and set ${\mathbf{C}}^{{\left( {\text{s + 1}} \right)}} = ( {{\mathbf{C}}^{{\left( {\text{s}} \right)}} \cup \{ {{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} } \}} ) - \{ {{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} } \}$, updating the flows matrix as:

  $${\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} = {\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }}$$
  $${\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} }} = {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} }} \quad \forall \ell \ne {\text{i}}, {\text{j}}$$
  $${\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} ,{\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} }} = {\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} }}$$

  and the travel time ${\text{t}}_{{{\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }}$ for each ${\text{C}}_{\ell }^{{\left( {\text{s}} \right)}} \in {\mathbf{C}}^{{\left( {\text{s}} \right)}}$ as:

  $${\text{t}}_{{{\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} ,{\text{C}}_{\ell }^{{\left( {\text{s}} \right)}} }} = \frac{{{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} }} }}{{{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} }} }}{\text{t}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + \frac{{{\text{T}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} }} }}{{{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} }} }}{\text{t}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} \quad {\text{if}}\ \hbox{max} \, \left\{ {{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} ,{\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {{\text{s}} + 1} \right)}} }} } \right\} \, > 0$$
  $${\text{t}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} = \frac{{{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} }} }}{{{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} , {\text{C}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} }} }}{\text{t}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} + \frac{{{\text{T}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} }} }}{{{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} , {\text{C}}^{{\left( {\text{s}} \right)}} }} + {\text{T}}_{{{\text{C}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} }} }}{\text{t}}_{{{\text{C}}_{\text{j}}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} \quad {\text{if}}\ \hbox{max} \, \left\{ {{\text{T}}_{{{\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} , {\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} }} , {\text{T}}_{{{\text{C}}_{\ell}^{{\left( {\text{s}} \right)}} , {\text{C}}_{\text{i}}^{{\left( {\text{s + 1}} \right)}} }} } \right\} \, = 0$$

  put n^(s+1) = n^(s) − 1, s = s + 1 and go to step 1.

Once the algorithm has finished in a sth stage, set K = n^(s), ${\text{C}}_{\text{i}} = {\text{C}}_{\text{i}}^{{\left( {\text{s}} \right)}}$ and ${\mathbf{C}} = {\mathbf{C}}^{{ ( {\text{s)}}}}$ and we obtain the set of initial configurations C = {C₁,…, C_K}.

Appendix 3: Initialization algorithm

The details of the algorithm are the followings:

• Step 0 (Fix the parameters) Fix α₁ > 1 and 0 < α₂ < 1, which allow us to select the configuration that will be the foci. Let n_p be the size of the initial population for the genetic algorithm to be generated.

• Step 1 (Calculate JR and SC) Starting from C = {C₁,…, C_K} the JR and the SC are calculated for each configuration:

  $${\text{JR}}_{\text{i}} = \frac{{{\text{T}}_{{{\mathbf{C}},{\text{C}}_{\text{i}} }} }}{{{\text{T}}_{{{\text{C}}_{\text{i}} ,{\mathbf{C}}}} }}\;{\text{and}}\;{\text{SC}}_{\text{i}} = \frac{{{\text{T}}_{{{\text{C}}_{\text{i}} , {\text{C}}_{\text{i}} }} }}{{{\text{T}}_{{{\text{C}}_{\text{i}} ,{\mathbf{C}}}} }}\quad {\text{for i}} = 1, \ldots ,{\text{K}}$$

• Step 2 (Build the n _p individuals)

  For s = 1, …, n_p do the following steps 2.1–2.3:

  • Step 2.1. (Generate α ₁ and α ₂ ) Draw $\alpha_{1}^{{ ( {\text{s)}}}}$ from a Unif(1,α₁) and $\alpha_{2}^{{ ( {\text{s)}}}}$ from a Unif(α₂,1) where Unif(a,b) a < b ∈ R denotes the uniform distribution in (a,b)

  • Step 2.2. (Determinate the foci) Build F ^(s) = {C_i: JR_i ≥ $\alpha_{1}^{{ ( {\text{s)}}}}$ or SC_i ≥ $\alpha_{2}^{{ ( {\text{s}})}}$}

  • Step 2.3. (Determinate the markets)

    For each C_i ∈ F ^(s) build:

    $${\text{M}}_{\text{i}}^{{ ( {\text{s)}}}} = \left\{ {{\text{Cj}} \in {\text{C}}:\prod ({\text{Ci}},{\text{Cj}}) = \max_{{{\text{C}}_{\text{k}} \in {\text{F}}^{{ ( {\text{s)}}}} }} \prod \left( {{\text{C}}_{\text{k}} , {\text{C}}_{\text{j}} } \right)} \right\}$$

    where Π(C_i,C_j) = $\frac{{{\text{T}}_{{{\text{C}}_{\text{i}} , {\text{C}}_{\text{j}} }}^{ 2} }}{{{\text{T}}_{{{\text{C}}_{\text{i}} ,{\mathbf{C}}}} {\text{T}}_{{{\mathbf{C}},{\text{C}}_{\text{j}} }} }} + \frac{{{\text{T}}_{{{\text{C}}_{\text{j}} , {\text{C}}_{\text{i}} }}^{ 2} }}{{{\text{T}}_{{{\text{C}}_{\text{i}} ,{\mathbf{C}}}} {\text{T}}_{{{\mathbf{C}},{\text{C}}_{\text{j}} }} }}$

As a result of the algorithm we obtain a set of n_p initial markets M ^(s) = $\{ {{\text{M}}_{\text{i}}^{{ ( {\text{s)}}}} ;{\text{i}}:{\text{C}}_{\text{i}} \in {\mathbf{F}}^{{ ( {\text{s)}}}} } \}$ s = 1,…, n_p that are taken as the initial population for the genetic algorithm.

Parameters α₁ and α₂ determine the minimum JR and SC values required for a BSU to be a potential focus. Given that JR and SC are not very high in Aragón, in our empirical example we have chosen α₁ = 1.3 and α₂ = 0.55.

Appendix 4: Partitioning algorithm

Let S = {S ^(p); p = 1,…, P} be the set of solutions obtained by the EA where S ^(p) = $\{ {{\text{S}}_{\ell }^{{ ( {\text{p)}}}} ;\ell = 1, \ldots ,{\text{n}}^{{ ( {\text{p)}}}} } \}$; p = 1,…, P and let ${\text{FS}}_{\ell }^{{ ( {\text{p)}}}}$ be the element of ${\text{S}}_{\ell }^{{ ( {\text{p)}}}}$ with larger population that will be named focus. We denote by M ^(s) = $\{ {{\text{M}}_{\text{i}}^{{ ( {\text{s)}}}} ;{\text{i}} = 1, \ldots ,{\text{m}}^{{ ( {\text{s)}}}} } \}$ the partition in LLMs obtained at the sth iteration of the algorithm and by F ^(s) = $\{ {{\text{F}}_{\text{i}}^{{ ( {\text{s)}}}} ;{\text{i}} = 1, \ldots ,{\text{m}}^{{ ( {\text{s)}}}} } \}$ the set of foci where $\text{F}_{i}^{(s)}$ is the focus of the market $\text{M}_{i}^{(s)}$.

The algorithm consists of the following steps:

• Step 0 (Initialization)

  Fix porc_min = 100 the minimum percentage of solutions that will have to include a given BSU in order to become eligible as a focus of a LLM and Δ the decrement of the values of porc_min. We denote LLM the set of LLMs partitions and SF the set of foci. Set LLM ⁽⁰⁾ = ∅ and SF ⁽⁰⁾ = ∅. Set t (number of iterations) equals to 1.

• Step 1 (Initial selection of foci)

  Calculate, for each B_i ∈ B, the percentage of solutions of S where B_i appears as a focus of a market that is given by:

  $${\text{porc}}_{\text{i}} = 100\frac{{{\text{Card }}\left( {\left\{ {{\text{p}} \in \{ 1, \ldots ,{\text{P}}\} :\exists \ell \in \{ 1, \ldots ,{\text{n}}^{{ ( {\text{p)}}}} \} {\text{ with FS}}_{\ell }^{{ ( {\text{p)}}}} = {\text{B}}_{\text{i}} } \right\}} \right)}}{\text{P}}$$

  Build F ⁽⁰⁾ = {B_i ∈ B : porc_i ≥ porc_min} and m⁽⁰⁾ = Card(F ⁽⁰⁾). Put s = 0.

• Step 2 (Delimitation of markets)

  For j = 1,…, m^(s) build:

  $${\text{M}}_{\text{j}}^{{ ( {\text{s)}}}} = \left\{{{\text{B}}_{\text{i}} \in {\mathbf{B}}:{\text{porc\_enc}}_{\text{i,j}}^{{ ( {\text{s)}}}} = \max_{{1 \, \le {\text{ k }} \le {\text{ m}}^{{ ( {\text{s)}}}} }} {\text{porc\_enc}}_{\text{i,k}}^{{ ( {\text{s)}}}} } \right\}$$

  where ${\text{porc\_enc}}_{\text{i,k}}^{{ ( {\text{s)}}}} = 100\frac{\text{Card}\left( \left \{ \text{p} \in \left \{\text{1,} \ldots , \text{P} \right \} : \exists \ell \in \left \{\text{1,} \ldots , \text{n}^{\text{(p)}} \right \} \ \text{with} \ \text{B}_{\text{i}} , {\text{F}}_{\text{k}}^{\text{(s)}} \in {\text{S}_{\ell }^{ \text{(p)}}} \right \} \right)}{\text{P}}$is the percentage of solutions of S in which B_i and the focus $F_{k}^{(s)}$ are placed within the same market. In case of ties, B_i will be assigned to the market whose focus is closer in terms of commuting time.

• Step 3 (Update of foci)

  Build F ^(s+1) = $\{ {\text{F}_{i}^{{\left( {s + 1} \right)}} ;i = 1, \ldots ,\text{m}^{(s)} } \}$ where ${\text{F}}_{\text{i}}^{{({\text{s}} + 1)}}$ is the BSU of the market ${\text{M}}_{\text{i}}^{{ ( {\text{s)}}}}$ with the largest resident occupied population. If F ^(s) = F ^(s+1), go to step 4. Otherwise set m^(s+1) = m^(s), put s = s + 1 and go to step 2.

• Step 4 (Validity of the solution)

  The validity of the solution M = M ^(s) is analyzed, i.e., it is checked if M satisfies (1)–(3). When these constraints are not satisfied, the algorithm stop. Otherwise the connectivity of the solution is analyzed. To that aim we consider for each BSU B_i its connection with the focus F of the market M ∈ M where the unit is placed, that is to say, we analyze the existence of a path {${\text{Path}}_{\ell }^{{\prime }}$; ℓ = 1,…, k} ⊆ M such that Path₁ = B_i, Path_k = F and Path_ℓ+1 ∈ L(Path_ℓ). When that path does not exist, B_i is reassigned to the market M′ ∈ M with focus F’ connected to B_i and such that the commuting time is minimum. This step is repeated until each B_i is placed in a market. If the new solution M′ satisfies (1)–(3) set LLM ^(t) = M′ the obtained solution and SF ^(t) = F′ the foci of markets in M′ and go to Step 5; otherwise, stop the algorithm.

• Step 5 (Decrease of porc_min)

  If LLM^(t − 1) ≠ LLM^(t) set LLM = LLM ∪ {LLM^(t)}, SF = SF ∪ {SF ^(t)}. Set porc_min = porc_min − Δ. If porc_min > 0, set t = t + 1 and go to step 1. Otherwise stop the algorithm.

The algorithm provides a ranking of foci SF = $\{ {{\mathbf{SF}}^{{\left( {{\text{t}}_{\text{i}} } \right)}} ; \, 1 \le t_{1} < t_{2} < \cdots < t_{m} } \}$ sorted by the values of porc_min where each focus was first selected. This information can be useful for planners in order to make decisions about the most appropriate locations for community, social and administrative services by taking into account the labor mobility patterns of the citizens.

In the empirical example, this algorithm was applied with Δ = 1 and stopped when the porc_min < 58 with m = 10 possible LLMs delimitations.

Other cluster methods can be applied.⁵ However, in our experience the above algorithm is quick and efficient.