Finding the maximum or longest increasing subsequence (LIS) within a given a set of numbers is a classic Computer Science problem. The algorithm is central to many tools and is used in various disciplines related to mathematics, physics etc. In the field of Bioinformatics, LIS is used to find DNA subsequences matching a certain criteria. By recognizing certain patterns in DNA sequences scientists can determine diseases, genetic disorders etc.

Problem

Given a sequence of N integers, find the longest strictly increasing subsequence. For example in the sequence 56, 21, 33, 22, 34, 78, 11, 35, 46, the longest increasing subsequence is 21, 33, 34, 35, 46. In this context we will consider a NXN matrix of integers and find a sequence which is simply a series of adjacent squares. A square may not be used more than once. In the following 10 X 10 grid:

  | 0  1  2  3  4  5  6  7  8  9
--+------------------------------
0 | 95 47 30 36 60 31 57 66 12 55
1 | 35 57 41 13 82 80 71 93 31 62
2 | 89 36 98 75 91 24 95 53 37 99
3 | 25 45 26 17 15 84 80 73 96 21
4 | 75 22 43 96 96 36 64 31 45 86
5 | 45 99 68 74 54 14 93 17 14 55
6 | 14 42 52 54 34 50 22 84 32 41
7 | 90 44 73 10 71 84 20 12 55 52
8 | 95 33 25 31 76 45 44 84 90 52
9 | 94 64 95 24 41 63 87 93 79 12

the longest increasing subsequence is of length 8 consisting of entries 24, 25, 33, 44, 52, 68, 74, 96 as highlighted in red.

Solution

Each number and its position in the sequence are stored in an array. In the first pass, neighbors for each item in the array are found and are stored along with the item. Items are then sorted by their descending value. In the second and final pass, weights are assigned to the neighbors. The neighbor with the least value is assigned weight 1, the next highest neighbor is assigned 2 and so on. To get the longest increasing subsequence, find the element with the highest weight and print its neighbors in the descending order of weights.

Unzip the attachment and cd into the newly created folder. Compile the classes run as follows:

C:\MaximumIncreasingSubsequence>javac -d bin src\*.java
 
C:\MaximumIncreasingSubsequence>java -classpath bin MaximumIncreasingSubsequence small-grid.txt
(24, 0) with value 129 (weight = 15)
(24, 1) with value 138 (weight = 14)
(25, 2) with value 167 (weight = 13)
(25, 3) with value 236 (weight = 12)
(25, 4) with value 300 (weight = 11)
(26, 4) with value 338 (weight = 10)
(27, 5) with value 438 (weight = 9)
(28, 4) with value 469 (weight = 8)
(27, 4) with value 596 (weight = 7)
(27, 3) with value 709 (weight = 6)
(26, 2) with value 723 (weight = 5)
(27, 1) with value 778 (weight = 4)
(28, 1) with value 866 (weight = 3)
(27, 0) with value 939 (weight = 2)
(26, 0) with value 953 (weight = 1)
Program took: 380 milliseconds

Attachments

Source code

Introduction

This article discusses how to install and configure Apache 2.0.x and Tomcat (or JBoss) to communicate with each other via mod_jk. This article also discusses how to setup a Tomcat cluster using mod_jk to balance work load.

Install JDK

1. Download JDK rpm and launch the executable by running the following commands:

   chmod a+x j2sdk-1_4_2_<version number>-linux-i586-rpm.bin
   ./j2sdk-1_4_2_<version number>-linux-i586-rpm.bin

2. Type yes and agree to the binary license agreement.
3. Become root by running su and entering root password
4. Install the rpm. This will install JDK at /usr/java/j2sdk-1.4.2:

   rpm -iv jdk-1.4.2_<version number>.i386.rpm

5. Edit PATH environment variable to include bin directory of JDK by adding the following lines to /etc/profile:

   JAVA_HOME=/usr/java/j2sdk-1.4.2_<version number>
   export JAVA_HOME
   PATH=$JAVA_HOME/bin:$PATH

Avoid Firewall and Routing issues

1. Disable the default firewall (unless you can configure it to allow multicast packets)

   /etc/init.d/iptables stop
   /etc/init.d/ip6tables stop

2. Add this multicast route:

   /sbin/route add -net 224.0.0.0 netmask 240.0.0.0 dev eth<id>

Install Apache

1. Download httpd-2.0.50-i686-pc-linux-gnu.tar.gz into a temporary location and run the following commands to install Apache at /usr/local/share:

   unzip < httpd-2.0.50-i686-pc-linux-gnu.tar.gz | tar xvf -
   cd httpd-2.0.50
   ./install-bindist.sh /usr/local/share/apache-2.0.50

2. Edit PATH environment variable to include bin directory of Apache by adding the following lines to /etc/profile:

   APACHE_HOME=/usr/local/share/apache-2.0.50
   export APACHE_HOME
   PATH=$APACHE_HOME/bin:$PATH
   export PATH

Install Tomcat

1. Move apache-tomcat-x.y.z.zip.zip to /usr/local/share (I will refer to the installation folder by this path) and unzip it:

   unzip apache-tomcat-x.y.z.zip

2. Edit PATH environment variable to include bin directory of Apache by adding the following lines to /etc/profile:

   TOMCAT_HOME=/usr/local/share/apache-tomcat-x.y.z
   export TOMCAT_HOME
   PATH=$TOMCAT_HOME/bin:$PATH
   export PATH

Configure mod_jk

1. Copy the mod_jk binary to $APACHE_HOME/modules directory and rename it as mod_jk.so
2. Add the following lines at the very bottom in $APACHE_HOME/conf/httpd.conf
   
   

   #Include mod_jk configuration file
   Include conf/mod-jk.conf

3. Under $APACHE_HOME/conf create mod-jk.conf and write the following lines in it:
   
   

   # Load mod_jk module
   # Specify the filename of the mod_jk lib
   LoadModule jk_module modules/mod_jk.so
   # Where to find workers.properties
   JkWorkersFile conf/workers.properties
   # Where to put jk logs
   JkLogFile logs/mod_jk.log
   # Set the jk log level [debug/error/info]
   JkLogLevel info
   # Select the log format
   JkLogStampFormat "[%a %b %d %H:%M:%S %Y]"
   # JkOptions indicates to send SSK KEY SIZE
   JkOptions +ForwardKeySize +ForwardURICompat -
   ForwardDirectories
   # JkRequestLogFormat
   JkRequestLogFormat "%w %V %T"
   JkMount /Context/* loadbalancer

   
   
   Where Context is the web-app context name.

4. Under $APACHE_HOME/conf create workers.properties and populate it as follows:
   
   

   # Worker list
   worker.list=loadbalancer
   # Define Node1
   worker.node1.port=8009
   worker.node1.host=node1.ip.address
   worker.node1.type=ajp13
   worker.node1.lbfactor=1
   #worker.node1.local_worker=1 (1)
   worker.node1.cachesize=10
   # Define Node2
   worker.node2.port=8009
   worker.node2.host=node2.ip.address
   worker.node2.type=ajp13
   worker.node2.lbfactor=1
   #worker.node2.local_worker=1 (1)
   worker.node2.cachesize=10
   # Load-balancer
   worker.loadbalancer.type=lb
   worker.loadbalancer.balanced_workers=node1, node2
   worker.loadbalancer.sticky_session=1
   worker.loadbalancer.local_worker_only=1

   Add one worker per node and name workers as node1, node2, node3 … nodeN. Don’t forget to define all these as load balancing workers:

   worker.loadbalancer.balanced_workers=node1, node2, node3, ... nodeN

   Replace node1.ip.address with IPAddress of Tomcat(or JBoss) instance1 and so on. The attribute sticky_session=1 tells mod_jk that SESSION ID’s should be routed back to the same Tomcat worker. The lbfactor determines the number of requests a worker receives. The higher the lbfactor the more requests a worker receives.

5. Each node should be named according to the names specified in workers.properties. On each Tomcat instance, edit $TOMCAT_HOME/conf/server.xml. Locate the <Engine> element and add an attribute called jvmRoute:

   <Engine name="Catalina" defaultHost="localhost" jvmRoute="node1">
   …
   </Engine>

NOTE: THE jvmRoute ATTRIBUTE MUST MATCH THE NAME SPECIFIED IN workers.properties.

Start Apache and Tomcat

1. Start Apache:

   $APACHE_HOME/bin/apachectl -k start

2. To stop use:

   $APACHE_HOME/bin/apachectl -k stop

3. Start Tomcat:

   $TOMCAT_HOME/bin/startup.sh

4. Stop Tomcat:

   $TOMCAT_HOME/bin/shutdown.sh

Introduction

This article discusses how to configure Apache and OpenSSL to serve https content. We assume Apache is installed and OpenSSL commands are available in the PATH environment variable. Otherwise install Apache and OpenSSL using yum or apt.

Create a working directory

Let’s call it sslcert:

$mkdir sslcert

Create two subdirectories under it

Create 2 subdirectories under sslcert. Let’s call them certs and private.

$cd sslcert
$mkdir certs private

Create a database to keep track of each certificate signed

Run the following commands:

$echo '01' > serial
$touch certindex.txt

Make a custom config file for openssl to use

Copy the config elements into a file called openssl.cnf. (edit as needed):

Boggle is a word game designed by Allan Turoff. The objective of the game is to construct as many different words as you can from a grid of letters. You can use letters (in positions) that are next to each other – above, below, left, right, diagonal but you can not use the same letter more than once. Each word must be defined in a dictionary. Points are assigned according to the length of the word.

The source code attached here is an implementation of this game in Java. It demonstrates recursion to solve problems like this game. The program attempts to construct words as defined in dic.txt from puzzle.txt. Points are assigned according to the length of the word.

Instructions

To run this program you need JDK 5 or better installed. You can get JDK from here.

1. Unzip the file. This creates a folder called Boggle
2. Compile the code: javac src\*.java -d bin
3. Run the program: java -classpath bin Boggle

C:\Boggle>javac src\*.java  -d bin
 
C:\Boggle>java -classpath bin Boggle
Enter boggle file:
puzzle.txt
Opened puzzle.txt
Enter dictionary name:
dic.txt
Opened dic.txt
Reading files...
Solving...
Word            | Points | Path
----------------+--------+-----------------------
            APR |      1 | [(0,0), (0,1), (0,2)]
      APRTOCFED |     15 | [(0,0), (0,1), (0,2), (1,2), (1,1), (1,0), (2,0), (2,1), (2,2)]
            COT |      1 | [(1,0), (1,1), (1,2)]
            FED |      1 | [(2,0), (2,1), (2,2)]

Attachments

Source code